ASTRONOMY 
LIBRARY 


A  TREATISE 

ON 

THE   SUN'S    RADIATION 


AND 


OTHER  SOLAR    PHENOMENA'' 


IN  CONTINUATION  OF  THE  METEOROLOGICAL   TREATISE  ON 

ATMOSPHERIC  CIRCULATION  AND  RADIATION,   1915 


BY 

FRANK  H.  BIGELOW,  M.A.,  L.H.D 

PROFESSOR  OF   METEOROLOGY  IN  THE  U.   S.   WEATHER  BUREAU,   iSpI-IQIO 
AND  IN  THE  ARGENTINE  METEOROLOGICAL  OFFICE  SINCE    IQIO 


FIRST  EDITION 


NEW  YORK 

JOHN  WILEY  &   SONS,  INC. 

LONDON:   CHAPMAN  &  HALL,  LIMITED 

1918 


*  •  #  *•  • 


Copyright,  1918,  by 
FRANK   H.    BIGELOW 


PUBLISHERS  PRINTING  COMPANY 
207-217'  West  Twenty-fifth  Street,  New  York 


PREFACE 

THERE  are  two  fundamental  problems  in  solar  physics  that 
need  an  immediate  solution:  (l)  the  nature  of  the  solar  radiation, 
whether  it  is  black  or  gray,  that  is,  whether  it  has  full  or  imper- 
fect efficiency,  together  with  its  amount  at  the  sun,  and  hi  the 
different  layers  of  the  earth's  atmosphere;  (2)  the  physical  con- 
ditions at  which  this  radiation  generates,  and  the  pressures, 
temperatures,  densities,  and  gas  efficiencies  that  are  concerned 
with  the  observed  phenomena  in  the  solar  spectra.  It  has 
been  found  that  the  system  of  non-adiabatic  meteorology,  which 
was  explained  in  the  Author's  Treatise  on  Atmospheric  Circula- 
tion and  Radiation  becomes  equally  applicable  in  the  solar 
atmospheres  by  means  of  a  simple  transformation.  These 
terms  have  been  computed  for  several  monatomic  gases,  by 
the  method  of  trials,  and  the  results  are  not  only  instructive  in 
many  directions,  but  they  are  in  complete  harmony  with  the 
observations  made  with  the  spectro-heliograph  and  the  spectro- 
bolometer.  The  radiation  is  black  at  5.85  gram  calories  per 
square  centimeter  per  minute,  when  reduced  to  the  equivalent 
at  the  distance  of  the  earth;  it  loses  1.87  calories  in  the  hemi- 
spherical non-adiabatic  shell  of  the  sun,  and  is  effective  on  the 
outside  of  the  earth's  atmosphere  at  3.98  calories;  the  course 
of  its  depletion  is  measured  by  the  bolometer  to  2.47  calories 
at  the  sea  level.  This  conforms  with  the  thermodynamic  con- 
ditions of  the  two  atmospheres.  The  pyrheliometer  fails  to 
record  three  important  depletions,  and  it,  therefore,  cannot 
account  for  more  than  1.94  calories  by  the  Bouguer  formula. 
The  following  Treatise  is  a  continuation  of  the  earlier  one  on 
Meteorology.  FRANK  H.  BIGELOW. 

SOLAR  AND  MAGNETIC  OBSERVATORY, 
PlLAR-CORDOBA  ARGENTINA. 
December,  1917. 


in 


380098 


TABLE  OF  CONTENTS 

«  PAGE 

PREFACE         .- 1  . iii 

CHAPTER  I 

THE  THERMODYNAMIC  PROCESSES  IN  THE  SOLAR  ATMOSPHERE  .     .     .  1 

Introduction       *     .      .     .„     .     .      .      .      .     .......  1 

Historical  Remarks .....  -w.     ,    '. 9 

Derivation  of  the  Non-Adiabatic  System  of  Equations      ....  10 

Preliminary  Summary  of  Conditions 13 

The  Working  Equations 15 

Adiabatic  and  Non-Adiabatic .      .  16 

The  Thermodynamic  Equation  for  the  Conservation  of  Energy  .      .  18 
The  Initial  Constants  and  Coefficients  for  the  Computations  in  the 

Solar  Atmosphere 19 

Notation  and  Fundamental  Formulas  for  the  Electromagnetic  Field  .  28 

Poynting's  Equation  for  the  Flux  of  Radiation 30 

Pressure  of  Radiation 32 

The  Mean  Flux  of  Energy       .      .      .      .'......      .      .  33 

The  Mean  Pressure  of  Radiation       ......      .     .     .      .      .  34 

Emission  and  Temperature,  Stefan's  Law 34 

Summary  of  the  Formulas  of  Radiation 35 

Tables  of  the  Constituents  of  Radiation  in  the  Volume     ....  37 

Definition  of  Terms  and  Dimensions 38 

Table  of  Astronomical  Constants  41 


CHAPTER   II 

COMPUTATION  OF  THE  THERMODYNAMIC  TERMS 43 

General  Remarks     . 43 

The  Atomic  and  Molecular  Weights .      .  56 

The  Distribution  of  the  Temperatures .58 

The  Heights  at  Which  the  Same  Temperature  Occurs  for  the  Differ- 
ent Elements  by  the  Hyperbolic  Law  . 62 

The  Constant  Temperature  Near  the  Photosphere  and  the  Variations 

of  the  Temperature  Outside  the  Isothermal  Region        ....  66 

The  Distribution  of  the  Pressures 68 

The  Distribution  of  the  Densities 74 

The  Gas  Efficiency  of  the  Different  Elements .      .  78 

The  Cause  of  the  Sharp  Limb  of  the  Sun 79 

v 


VI  TABLE    OF    CONTENTS 

CHAPTER    III  PAGE 

THE  DETERMINATION  OF  THE  "SOLAR  CONSTANT"  OF  RADIATION  IN  THE 

ISOTHERMAL  LAYER  OF  THE  SUN 83 

Statement  of  the  Problem 83 

The  Distribution  of  the  Free  Heat  (Qi  —  Qo) 85 

The  Distribution  of  the  Entropy  (Si— So) 88 

The  Distribution  of  the  Work  of  Expansion  (Wi  —  Wo)      ....     91 

The  Distribution  of  the  Inner  Energy  (£/i-t/o) 92 

The  Distribution  of  the  Radiation  Potential  KIQ 95 

Method  of  Computing  the  Coefficients  and  Exponents  in  the  Formula 
of  Radiation   .      .      .      .     ,     .     ....     ...      .      .      .      .     96 

The  Distribution  of  the  Exponent  (a)  in  Kw  =  c  Ta 100 

The  Distribution  of  the  Coefficient  (log  c) .101 

The  Solar  Constant  of  Radiation 103 

The  Mean  Values  of  the  Coefficient  and  the  Exponent  of  Radiation 

in  the  Stefan  Law 104 

The  Logarithmic  Spiral  Formulas 109 

The  Logarithmic  Spiral  for  a =35°  10' Ill 

The  Solar  True  Radiation  at  the  Distance  of  the  Earth    .      .      .      .115 

The  Average  Physical  Conditions  in  the  Solar  Strata  Where  the  Radi- 
ation Originates   .      .'     .     .      .     »"    .      .      .     .     .      .     .      .      .   116 

The  Evaluation  of  Ja  =  Ci  TV-Co  TV0  in  gr.  cal./cm.2  min.  .      .      .117 

CHAPTER   IV 
THE  COEFFICIENTS  IN  THE  STEFAN  AND  THE  WIEN-PLANCK  FORMULAS 

FOR  BLACK-BODY  RADIATION ,.     .   120 

The  Conversion  from  the  (M.K.S.)  to  the  (C.G.S.)  System    .      .      .   120 
The  Coefficients  in  the  Wien-Planck  Formula  of  Spectrum  Radiation  127 

The  Formulas  of  Computation 129 

The  Thermodynamics  of  Radiation  in  the  Solar  Atmospheres      .      .   134 

Tables  for  the  Mean  Square  Velocity 135 

Tables  for  the  Number  of  Molecules  per  Cu.  Cm .136 

Tables  for  the  Thermal  Coefficient  in  P  V=K  T    .      .      .     .     .      .   138 

Tables  for  the  Number  of  Molecules  in  One  Gram       .....   139 

Tables  for  the  Mean  Kinetic  Energy  of  Motion      .      .      .      .      .      .141 

.   Tables  for  the  Boltzmann's  Entropy  Coefficient 142 

Tables  for  the  Planck's  Wirkungsquantum 146 

Tables  for  the  Wien-Planck  Coefficients  ci,  c2     .      ....      149,151 

Tables  for  the  Volume  Intensity  Coefficient       .  153 

Tables  for  the  Kinetic  Energy  per  Unit  Volume      .      .      .      .      .      .   155 

Check  on  the  Computations 160 

Second  Computation  of  the  Wien-Planck  Coefficients        .      .      .      .161 

CHAPTER  V 

THE  ELEMENTS  OF  BLACK  RADIATION  IN  THE  ATMOSPHERES  OF  THE  SUN 

AND  THE  EARTH.     .   '*.    «~    • -^8 

The  Concentrations  of  Black  Radiation  in  Gaseous  Media     .      .      .    168 


TABLE    OF    CONTENTS  Vll 

THE   ELEMENTS  OF   BLACK   RADIATION   IN   THE  ATMOSPHERES  OF  THE 

SUN  AND  THE  EARTH — Continued  PAGE 

Tables  for  the  Specific  Volume  Density  .      .      .      .      ...»      .  171 

Tables  for  the  Total  Pressure  per  Unit  Volume       .      .      .      .      .      .  175 

Tables  for  the  Energy  of  Black  Radiation    .      .      ,    :.     .'-    .     .      .  177 

Tables  for  the  Mechanical  Force 179 

Tables  for  the  Specific  Entropy .  181 

Tables  for  the  Specific  Intensity  of  the  Entropy  Radiation     .      .      .  183 

Tables  for  the  Unpolarized  Specific  Intensity 185 

Tables  for  the  Total  Volume  Inner  Kinetic  Energy      .   . ..      ,      .      .  186 

Certain  Data  in  the  Radiation  Terms     .      ......      .      .      .  187 

Tables  for  the  Thermodynamic  Pressure  of  Kinetic  Energy    .      .      .  188 

The  Discontinuity  at  the  Levels  Where  Radiation  Is  Generated        .  191 

CHAPTER  VI 
RECONCILIATION  OF  THE  DATA  DERIVED  FROM  THE  PYRHELIOMETER  AND 

THE  BOLOMETER 192 

The  Poynting  Surface-Flux  and  Volume-Density  Equation     .      .      .    192 
The  Cause  of  the  Change  from  the  Total  Solar  Radiation  at  5.85 
Calories  to  the  Effective  Radiation  3.98  Calories  Received  on  the 

Outer  Strata  of  the  Earth's  Atmosphere 195 

Summary  of  the  Terrestrial  Thermodynamic  Data 200 

The  Thermodynamic  Data  in  the  1000-Meter  Levels  .      .      ;      .      .  206 

Resume  of  the  Preceding  Results 210 

The  Potential  Energy  of  the  Solar  Radiation  in  the  Sun's  Atmosphere .  211 
The   Potential   Energy   of    the    Solar   Radiation  in    the    Earth's 

Atmosphere 214 

The  Third  Thermodynamic  Depletion     .      .      .      .'   .  '  _.      .     ,.      .  216 
The  Scattered  and  the  Absorbed  Radiation        .......  216 

The  Pyrheliometer  and  the  Bolometer  Observations 217 

The  Depletion  of  the  Solar  Radiation  from  5.85  Calories  in  the  Iso- 
thermal Layers  of  the  Sun  to  1.50  Calories  at  the  Sea  Level  of  the 

Earth 218 

The  Line  and  Band  Absorption  and  the  Scattering 222 

Direct  Readings  of  the  Pyrheliometers  at  Great  Heights  .      .      .      .   225 
Summary  of  the  Pyrheliometer  Results  as  Reduced  by  Bigelow's 

Method 226 

Computation  of  the  Pyrheliometer  Data  Leading  to  3.98  gr.  cal./cm.2 

min.  as  the  Intensity  of  the  Solar  Radiation  at  the  Earth  .      .      .   227 
Synchronism  of  the  Solar  and  the  Terrestrial  Variations  During  the 

Interval  1900-1915  in  Argentina .   231 

Short- and  Long-Range  Forecasts !/    .      ..     .      .  236 

CHAPTER  VII 

OTHER  SOLAR  PHENOMENA .     .     .     .     .     .  240 

Restatement  of  the  General  Line  of  Argument 240 

The  Effect  of  the  Solar  Isothermal  Shell  upon  the  Visible  Surface 
Phenomena  .  242 


Vlll  TABLE    OF    CONTENTS 

OTHER  SOLAR  PHENOMENA — Continued  PAGE 

The  Granulations,  Faculse,  Flocculi,  and  Prominences       ....  243 

Granulations 249 

Faculse,  Flocculi,  Prominences 250 

The  Origin  of  the  Sun-Spots 251 

The  Circulation  in  a  Solar  Vortex  or  Sun-spot         257 

The  Inward  and  Outward  Velocities  in  the  Solar  Vortices       .      .      .261 

The  Invisible  Deep-seated  Thermodynamic  Processes        ....  262 

The  General  Circulation  of  the  Sun  in  Latitude 263 

The  General  Circulation  of  the  Sun  in  Longitude 267 

The  Chromosphere  and  the  Inner  Corona     .      .  - 268 

The  Outer  Solar  Corona     .      .     .......      ....      >      .      .269 

Solar  Magnetism .      .      .      .      .  271 

The  Polar  Rays  of  the  Solar  Coronas  as  Evidence  of  a  Magnetized 

Sphere 273 

The  Zeeman  Effect  in  the  Sun's  Atmosphere      .......  275 

The  Distribution  of  the  Solar  Magnetism  as  Determined  by  the  Zee- 
man  Effect 277 

The  Solar  Spectra 280 

The  General  Solar  Spectrum 281 

Atmospheric  Refraction  and  Scattering  .      .      .   "  .      .      .      .      .      .  285 

The  Formulas  of  Refraction .      .      ...  286 

The  Density  p  as  Computed  by  the  Non-adiabatic,  the  Adiabatic,  and 

the  Bessel  Formulas 289 

The  Atmospheric  Transmission  for  Different  Wave-lengths  pk,  Col- 
lected According  to  the  Pyrheliometer  pw 290 

Results  for  Washington,  Mt.  Wilson,  and  Mt.  Whitney     .    291,-  292,  293 

The  Terrestrial  Values  of  the  Index  of  Refraction  .      .      .      .  '   .      .  293 

The  Solar  Values  of  the  Index  of  Refraction 294 

General  Remarks 295 

Variation  of  the  Intensity  of  the  Sun's  Radiation  in  Longitude    .      .  300 

CHAPTER  VIII 

THREE  THEORIES  OF  RADIATION 304 

The  Derivation    of   the   Wien-Planck  Formula  for    Black    Radia- 
tion in  a  Spectrum ...      .      .  305 

The  Derivation  of  Bigelow's  Functions  for  the   Potential  h     .      .  307 

Additional  Formulas .....      .      .  310 

The  Variable  k  in  all  Atmospheres  .      .  ' 312 

The  Atmospheric  Pressure .  313 

The  Variable  Potential  Coefficient  h 313 

The  Solar  Volume  Density  of  Radiation 314 

The  Terrestrial  Volume  Density  of  Radiation 315 

Evaluation  of  lr-7;;)      and  1 7-^1         318 

\k  TJB  \k  i/ p 

The  Variations  in  the  Wien  Displacement  Law     .•'...     .     .     .      .  319 

Another  Formula  for  the  Quantity  hs  .........  320 


TABLE    OF    CONTENTS  IX 

THREE  THEORIES  OF  RADIATION — Continued  PAGE 
The  Kinetic  and  the  Potential  Energies  in  Radiation,  Determined 

from  Thermodynamics  .  , .321 

The  Molecular  Potential  Energy  in  the  Earth's  Atmosphere  .  .  325 

The  Molecular  Potential  Energy  in  the  Sun 326 

Summary  .  .  .  .  .'..... 328 

Derivation  of  Certain  Formulas  for  Radiation,  lonization,  and 

Atmospheric  Electricity 329 

lonization,  Potential  Energy,  and  Frequency  .  ......  .  330 

Atmospheric  Electricity  .  .  .  ... 331 

The  Terrestrial  Atmospheric  Electricity  .  .  .  '  .  .  .  .  .  333 

The  Solar  Atmospheric  Electricity .  .  339 

Atmospheric  Electricity  and  the  Diurnal  Convection  .  .  .  .  342 

The  Fundamental  Quantities  in  Meteorology  and  in  Astrophysics  .  345 

Practical  Series  of  Thermodynamic  Terms 348 

The  Relations  Between  the  Kinetic  and  the  Potential  Energies  in 

Orbital  Oscillations  .  ...  ... 349 

Bohr's  Theory  of  Non-Radiating  Orbits  in  Atoms  .  .  .  .  .  350 

The  Series  of  Spectral  Lines t  .  .  .  .  .  .  .  351 

Derivation  of  the  Orbital  Formulas  ...  .  '  .  .  ' .  .  .  352 

Moseley's  Law,  Bigelow  Form  and  Millikan  Form 355 

Evaluation  of  I—, „• 357 

\Tl2          T22/ 

The  Electronic  Orbits  in  the  K  and  L  Series  of  Radiation  Lines 

for /*- Variable     .      .../'.' '.  ^.      .      .      .  358 

Ka  —  Radiation     .      .      .      . .»    .      .      .      .      .      .      '.     .      ,     .      .  359 

La  —  Radiation ,.*....  361 

The  Interpenetration  of  the  Electron  Orbits  at  the  Contact  of 

CoUision        ...     .      .     .     ,.. \  362 

The   Electromagnetic   Waves   Due   to   the   Sudden    Motion   and 

Stoppage  of  an  Electric  Charge  in  Collisions 363 

The  Variable  Intensity  of  the  Sun's  Radiation  in  the  26.68- Day 

Period  of  the  Synodic  Rotation ,  365 

International  Character  Numbers    .      ..     .    ^.      .      .      .     ..      .      .  366 

Synchronism  Between  the  Solar  Radiation  Intensity  and  the 
Terrestrial  Magnetic  Variations  in  the  365-Day  and  the  26.68- 
Day  Periods 368 

The  Effect  of  Dust  in  the  Lower  Strata      .      .      .      .....      .  369 

The  Systems  of  Units  Employed  in  Meteorology  . 371 

The  Bar  as  a  Practical  Unit  for  Absolute  Pressure 373 

Long  Range  Forecasts  of  Weather  Conditions       .      .      .      .      .      .  374 

Appendix.     The  Pyrheliometer  and  the  Poynting  Equation       .      .  375 


A  TREATISE  ON  THE  SUN'S   RADIATION 
AND   OTHER  SOLAR   PHENOMENA 

CHAPTER  I 
The  Thermodynamic  Processes  in  the  Solar  Atmosphere 

Introduction 

THIS  Treatise  on  Solar  Physics  is  the  direct  continuation  of  the 
author's  Meteorological  Treatise  on  the  Circulation  and  Radia- 
tion in  the  Atmospheres  of  the  Earth  and  the  Sun,  1915.  In  the 
earlier  work  the  fundamental  principles  and  the  formulas  were  suf- 
ficiently explained,  so  that  it  is  unnecessary  to  repeat  them  here. 
In  this  research  there  are  several  important  points  essential  to 
the  discussion  that  should  be  kept  in  mind  by  the  reader. 

(1)  In  the  Boyle-Gay  Lussac  Law,  P  =  p  R  T,  aU  the  terms, 
including  the  gas  efficiency  R,  are  variable.    This  is  the  general, 
non-adiabatic  case,  which  alone  is  applicable  to  free  atmospheres, 
the  adiabatic  case  being  only  occasionally  developed.    This 
change  of  Ra  =  constant  to  R  =  variable  is  very  revolutionary, 
because  it  carries  with  it  the  so-called  constants,  k,  h,  Ci,  c2,  in 
the  Wien-Planck  spectrum  formula,  and  a,  cr,  in  the  general 
Stefan  formula  for  black  radiation. 

(2)  The  transition  from  terrestrial  values  to  initial  solar 
values  of  Pv  =  R  T  is  effected  by, 

G^         gravity  acceleration  at  the  sun 
#0        gravity  acceleration  at  the  earth 
so  that  y  P  .  yv  =  y  R  .y  T,  at  one  point  on  the  sun. 

From  this  point,  by  the  method  of  trials  with  the  dynamic  and 
the  gravity  equations,  the  complete  thermodynamic  system  has 
been  worked  out  for  several  gases,  from  hydrogen,  monatomic, 
Hi  =  1.00  and  diatomic,  H2  =  2.00,  to  mercury  Hg  =  198. 
Each  gas  develops  independently  of  all  the  others,  and  possesses 

1 


ylLH  A   ^EATISE   ON   THE   SUN'S   RADIATION 

a  low  level  adiabatic  base  below  the  solar  photosphere,  a  middle 
isothermal  region  containing  the  photosphere,  and  a  non-adia- 
batic  region  extending  to  the  vanishing  planes  at  different  heights, 
H2  at  25,000  kilometers  above  the  photosphere,  He  at  11,000, 
carbon  (assumed  monatomic)  at  4,000,  calcium  at  1,000,  zinc  at 
675,  cadmium  at  380,  mercury  at  210  kilometers.  The  sharp 
disk  of  the  sun  is  not  an  optical  effect,  but  the  vanishing  level 
of  the  heavy  metallic  vapors  under  the  operation  of  the  prevailing 
thermodynamic  conditions.  The  computed  system  is  in  ad- 
mirable harmony  with  that  which  is  indicated  by  the  spectro- 
heliographic  observations. 

(3)  The  equation  of  condition  for  radiation  is  derived  directly 
from  the  first  law  of  thermodynamics,  dQ  =  dW  +  d  U,  by 
substituting  d  W  =  P  dv, 

*~-  T)  _    >.  rl 1  . 

•*    10    —  —   -tVlO   —    L  J.     . 

fll  —    VQ  Vi  —    VQ 

The    dimension    of    each     term     is     easily     reduced     to 
—^—-  .    —  —  T~jT2 J  >  tne  mean  kinetic  energy  per  unit  volume, 

and  it  is  so  described  by  Boltzmann,  Heaviside,  Lorentz,  W. 
Wien,  Planck,  A.  L.  Day  and  C.  E.  van  Ostrand,  Richardson, 
and  others  generally.  When  the  volume  energy  is  transmitted 
with  the  velocity  c  the  result  is  radiant  energy  or  flux, 

r  M      L     M-\ 

\_Y~T2'     ~T  ~  T3  r  conversion  factor  of   radiation-den- 

sity from  (M.  K.  S.)  to  (C.  G.  S.)  is  10;   that  of  the  radiant- 

Dplyin 

Joule 


flux  is  1,000.     Applying  to  a  =  -  —  j—  ,  the  dimensions  give: 


K  S--  '          - 

m.  sec.2  deg.4      m.3  deg.4'  '  cm.  sec.2  deg.4      cm.3  deg.4' 

(C.  G.  S.) 
Apply  ing.  to  a  =  a.  —,  the  dimensions  give: 

kilog.  J°ule          (M  K  S)-         gr' 

sec.3  deg.4       m.2  sec.  deg.4'  '  sec.3  deg.4 

erg.  s          < 


cm.2  sec.  deg. 


THERMODYNAMIC   PROCESSES   IN   SOLAR  ATMOSPHERE  3 

The  earlier  treatise  was  concerned  exclusively  with  the  volume 
density  equations  of  ponderable  gases,  and  not  with  the  flux 
equation.  This  treatise  passes  from  the  volume  density  of 
the  kinetic  energy  of  gases  to  the  electromagnetic  flux  of  radiation 
in  the  pure  aether,  and  to  its  source  near  the  bottom  of  the 
isothermal  layers  in  the  solar  envelope.  In  this  process  the 
Poynting  Equation  for  surface-flux  and  volume  density  is 
employed. 

(4)  In  computing  the  Boltzmann  entropy  coefficient,  k  = 

/(""        wi  7? 

-jy.  =  ~y»r~,  where  N  =  the  number  of  #-atoms  per  unit  mass, 

it  is  evident  that  while  in  the  adiabatic  system  k,  K,  N,  are  all 
constants,  k,  K,  are  certainly  variable  in  the  non-adiabatic 
system.  If  H  is  the  inner  molecular  kinetic  energy  per  unit 

3 

volume,  H  =  —  P  =  U,  in  monatomic  gases,  and  EQ  is  the  in- 
trinsic kinetic  energy  of  each  molecule,  assumed  in  the  usual 
kinetic  theory  of  gases  to  be  a  universal  constant,  we  have  for 

TJ  op 

the  number  of  molecules  per  unit  volume,  n  =  TT  =  TTTT,  and 

JbQ      z  £LQ 

thence  N  =  n  —  —  — ,  where  mH  is  the  mass  of  the  standard 
P        WH 

hydrogen  molecule.  It  appears  from  the  computations  that  N 
does  not  remain  constant,  but  diminishes  with  the  height, 
especially  in  the  very  rarefied  high-level  strata.  We,  therefore, 
encounter  a  dilemma,  because  either  N  or  EQ  is  variable,  and 

they  may  both  be  variables.     If  it  is  thought  that  N  =  — 

nifj 

is  necessarily  constant,  it  assumes  a  certain  view  regarding  mass, 
which  is  not  yet  settled  in  the  analytic  discussions.  If  E0  is 
made  a  variable  it  modifies  the  foundations  of  the  kinetic  theory 
of  gases.  Finally,  if  EQ  is  a  variable,  diminishing  with  the  height, 
the  law  is  unknown,  and  k,  h,  Ci,  c*,  become  different  variables 
from  those  herewith  computed  in  the  first  case.  We  have,  ac- 
cordingly, retained  E0  =  constant  and  allowed  N  to  become 
variable,  in  the  preliminary  discussion  of  gaseous  radiation. 


The  system  has,  also,  been  given  for  N  =  constant  and  E0  = 
variable. 

(5)  The  computations  have  led   to  the  following  important 
result  regarding  the  value  of  the  solar  constant  of  radiation. 
In  each  gas,  at  its  special  depth  below  the  photosphere,  there  is  a 
sudden  change  in  the  value  of  all  the  variable  coefficients  entering 
into  the  process  of  radiation.     The  mean  temperature  in  these 
layers  of  radiation  is  7655°  A.  absolute;    the  mean  logarithmic 
value  of  a  in  the  Stefan  law,  JQ  =  a  Ta  is  log  a  =  —5.74000. 
and  the  exponent  a  =  4.00;    so  that  JQ  at  the  distance  of  the 

srr  cal 

earth  is  equivalent  to  5.854  '- — r-.     This  agrees  with  the 

cm.2  nun. 

predicted  value,  Fig.  54,  preceding  volume;  the  terrestrial 
thermodynamics  established  the  effective  radiation  reaching  the 
earth  as. 3.980  calories,  equivalent  to  about  6950°  A.;  the  ob- 
served depletion  by  the  bolometer  continues  from  station  to 
station  down  to  Washington,  2.47  calories.  The  scattering 
process  is  apparently  equivalent  to  1.87  calories  in  the  solar 
atmosphere,  0.76  calorie  in  the  high  levels  of  the  earth's  atmo- 
sphere, and  0.25  to  0.54  calorie  in  the  low  levels;  the  absorption 
process  in  the  earth's  atmosphere  increases  to  about  0.120  calorie; 
the  free  heat  at  sea  level  is  1.50  calories  by  the  pyrheliometer. 
Abbot's  extrapolated  value  of  the  solar  constant,  1.95  calories,  is 
about  one-third  of  the  true  solar  constant.  He  has,  therefore, 
made  an  error  in  supposing  that  the  sun,  though  having  high 
temperatures,  radiates  with  very  low  efficiency.  On  the  con- 
trary, the  data  of  the  computations  on  solar  thermodynamics, 
and  those  on  the  terrestrial  thermodynamics,  both  agree  with 
the  results  of  the  bolometer  in  making  the  solar  radiation  of 
full  efficiency. 

(6)  More  specifically,  the  process  of  depletion  of  the  solar 
intensity  of  radiation  from  5.85  gr.  cal./cm.2  min.  to  1.30-1.50 
calories  at  the  sea  level  of  the  earth's  atmosphere  proceeds  by 
the  following  steps. 

a.  Temperature  of  origin  for  all  the  gases  =  7655°  absolute 
A.  in  a  deep  isothermal  layer. 

b.  Exponent  in  the  Stefan  Law  of  black  radiation  =  4.00. 


THERMODYNAMJC   PROCESSES   IN   SOLAR  ATMOSPHERE 


c.  Equivalent  intensity  at  the  mean  distance  of  the  earth  = 
5.85  calories. 

d.  The  depletion  in  the  hemispherical  shell  of  the  sun's 
atmosphere,   including  the  isothermal  and   the  non-adiabatic 
layers,  is  measured  by  the  decrease  in  brightness  of  the  solar 
disk  between  the  center  and  the  limb.    Abbot's  Table  55,  Vol. 
III.,  Ann.  S.  I.,  gives   the   bolometer    intensities   for  several 
spectrum  lines,  in  parts  of  the  intensity  at  the  center,  for  certain 
points  of  the  radial  distance.     Factors  are  computed  for  each 
line  showing  its  contribution  when  referred  to  an  undepleted 
intensity  from  the  center  to  the  limb.    The  following  summary 
gives  the  ordinates  computed  for  black  radiation  at  the  tem- 
peratures 7655°,   6950°,   5810°,   5456°,   and  Abbot's  observed 
ordinates  as  extrapolated  by  him  to  the  outside  of  the  earth's 
atmosphere,  marked  Total,  at  Mt.  Whitney,  Mt.  Wilson,  and 
Washington.    The  corresponding  spectra  are  in  relative  num- 
bers, but  they  can  be  reduced  to  calories  by  the  general  factor 
20.9: 

COURSE  OF  DEPLETION  OF  THE  SOLAR  RADIATION  DOWN  TO  THE  SEA  LEVEL 


SOLAR 

BOLOMETER 

PYRHELI- 

THERMODYNAMICS 

OBSERVATIONS 

OMETER 

Wave 
Lengths 

T 
7655° 
Black 

Factor 
of 
Deple- 
tion 

Effec- 
tive of 
Sun 
to 
Earth 

T 
6950° 
Black 

Total 
as 
Extra- 
polated 

Mt. 

Whit- 
ney 

Mt. 

Wilson 

Wash- 
ington 

Total 
as 
Extra- 
polated 
5810° 

Sea 
Level 
Gener- 
ally 
5450° 

0.323M 

9.790 

0.580 

5.678 

5.406 

1.500 

0.828 

0.764 

1.578 

0.932 

.386 

10.552 

.589 

6.215 

6.348 

3.690 

2.959 

2.733 

1.728 

2.218 

1.453 

.433 

10.294 

.633 

6.516 

6.514 

5.472 

4.413 

4.265 

3.370 

2.586 

.790 

.456 

9.884 

.661 

6.533 

6.454 

6.051 

4.992 

4.883 

3.918 

2.618 

.824 

.481 

9.411 

.679 

6.390 

6.268 

6.056 

5.254 

5.063 

4.118 

2.656 

.885 

.501 

9.031 

.690 

6.231 

6.116 

6.052 

5.445 

5.195 

4.268 

2.684 

.931 

.534 

8.337 

.709 

5.911 

5.946 

5.768 

5.267 

5.017 

4.202 

2.648 

.942 

.604 

6.893 

.738 

5.091 

4.959 

4.994 

4.670 

4.454 

3.833 

2.480 

.884 

.670 

5.721 

.762 

4.359 

4.217 

4.070 

3.856 

3.758 

3.294 

2.229 

.732 

.699 

5.188 

.770 

3.995 

3.664 

3.664 

3.495 

3.444 

3.063 

2.116 

.663 

.866 

3.191 

.806 

2.572 

2.511 

2.403 

2.471 

2.451 

2.208 

1.602 

1.302 

1.031 

1.994 

.830 

1.655 

1.603 

1.536 

1.748 

1.744 

1.574 

1.162 

0.970 

1.225 

1.199 

.845 

1.013 

0.986 

0.985 

1.065 

1.055 

1.000 

0.732 

.626 

1.655 

0.459 

.884 

0.406 

.389 

.466 

0.456 

0.410 

0.440 

.256 

.226 

2.097 

.204 

.899 

.183 

.176 

.211 

.201 

.191 

.181 

.130 

.117 

Calories 

5.85 

3.98 

3.98 

3.22 

2.96 

2.87 

2.47 

1.94 

1.50 

6  A  TREATISE   ON  THE   SUN'S  RADIATION 

The  solar  radiation  arrives  at  the  earth  as  black  and  of  an 
equivalent  temperature  6950°. 

e.  The  terrestrial  thermodynamics  reproduces  the  same  co- 
efficient and  exponent  in  the  Stefan  Law  as  correspond  with 
black  body  radiation  at  6950°  on  the  levels  60000  to  65000 
meters. 

/.  The  summary  of  the  several  thermodynamic  terms  in  the 
earth's  atmosphere,  from  the  sea  level  to  the  vanishing  plane, 
derived  directly  from  the  prevailing  temperatures,  which  is  the 
precise  equivalent  of  the  effective  radiation,  whatever  may  be 
the  details  of  the  paths  of  the  radiant  energy,  gives  the  following 
results: 

1.  Summation  of  the  free  heat 4.08  gr.  cal./cm2  min. 

2.  Summation  of  the  hydrostatic  pres- 

sure     4.08      " 

3.  Summation  of  the   inner  energy  and 

work 4.08      " 

4.  Summation  of  the  black  radiation.  .  .  3.94 

5.  The  free  heat  in  the  stratum 3.92      " 

6.  The    hydrostatic    pressure    in    the 

stratum 4.00  "      * 

7.  The  inner  energy  in  the  stratum 3.92  " 

8.  The  external  work  and  gas  efficiency.  4.02  " 

9.  The   kinetic   and  potential   energies 

with  the  absorbed  radiation 3.90      " 

g.  The  energy  of  the  solar  radiation  in  the  atmosphere 
separates  itself  into  the  following  terms: 

I.  The  kinetic  energy  for  temperature  effects. 
II.  The  potential  energy  =  0.641  kinetic  energy. 

III.  The  specific  heat  change  from  constant  volume  to  con- 

stant pressure,  or  from  the  inner  energy  of  the  Boyle- 
Gay  Lussac  Law  to  the  external  energy  due  to  the 
impressed  gravitation. 

IV.  The  energy  lost  by  non-selective  scattering. 
V.  The  energy  lost  by  selective  absorption. 

VI.  The  energy  transformed  into  ionization  products,  free 
electric  charges,  and  magnetic  fields. 


THERMODYNAMIC   PROCESSES   IN   SOLAR  ATMOSPHERE  7 

h.  The  effective  intensity  3.98  calories  minus  (III  +  IV  + 
V  +  VI)  gives  the  bolometer  ordinates  approximately.  The 
further  subtraction  of  the  potential  energy  II  gives  the  data 
observed  by  the  pyrheliometer  in  bulk,  as  the  kinetic  energy 
which  produces  its  temperature.  Hence,  the  proper  reduction 
of  the  pyrheliometer  observations  follows: 

H  =  the  kinetic  energy,  observed  by  the  pyrheli- 
ometer in  the  zenith. 
/  =  the  potential  energy   =  0.641  H. 
k  (H  +  /)  =  (100  -  p)    (H  +  J)  =  the  proportional  part 

lost  by  scattering. 
A  Ue  =  0.012  (e  —  eQ) j  a  small  correction  for  vapor 

pressure. 
a  —  the  nearly  constant  line  and  band  absorption 

in  the  spectrum. 
R  =  the  nearly  constant  specific  heat  term  at  the 

station. 
S  =  the  effective  solar  constant  outside  the  earth's 

atmosphere. 
S=  (H  +  7)  +  k  (H  +  J)  +  A  Ue  +  (a  +  R)  = 

3.98  calories. 

These  reductions  of  the  pyrheliometer  observations  agree 
at  many  stations  in  producing  3.98  calories.  This  refers  to  La 
Confianza  (4485  m.),  Mt.  Whitney  (4420),  La  Quiaca  (3465), 
Mt.  Wilson  (1780),  Bassour  (1160),  Mt.  Weather  (526),  Cordoba 
(438),  Pilar  (340),  Washington,  D.  C.  (34).  The  Bouguer  for- 
mula of  depletion  is  used  to  compute  p  and  /io  =  H  in  the 
zenith,  but  it  does  not  employ  the  objectionable  Langley-Abbot 
method  of  extrapolation  beyond  its  natural  limits.  Each  station 
produces  3.98  calories  with  variations  on  its  own  level. 

(7)  The  most  interesting  and  striking  result  is  that  the  solar 
radiation  originates  in  a  per  saltum  process,  resembling  a  radio- 
active discharge,  due  to  a  readjustment  of  the  electrons  and  to 
thermal  collisions.  In  the  case  of  hydrogen  it  was  necessary  to 
treat  it  as  a  monatomic  element,  HI  =  1.00,  below  the  radiation 
level,  but  as  a  diatomic  element,  HZ  =  2.00,  above  that  level, 
in  order  to  terminate  on  the  observed  vanishing  plane,  namely, 


8  A    TREATISE   ON   THE    SUN'S    RADIATION 

the  top  of  the  inner  corona  at  25000  kilometers.  This  is  direct 
evidence  of  dissociation  below,  and  association  above  the  plane 
where  the  radiation  originates.  That  plane  is  just  above  the 
true  adiabatic  strata  where  the  isothermal  layer  begins,  and 
at  the  levels  where  the  temperature,  the  pressure,  the  density, 
and  the  gas  coefficient,  all  suffer  very  quick  changes  of  gradient. 
The  factor  of  change  in  the  coefficients  of  radiation  is  of  the 
order  10~6,  dropping  from  a  very  high  tension  to  a  very  low 
tension  in  a  short  vertical  distance.  The  effect  of  this  phenom- 
enon, as  well  as  the  general  variability  of  the  coefficients  of 
radiation,  upon  the  theories  of  radiation  now  under  discussion, 
cannot  possibly  be  indicated.  It  has  been  our  purpose  to  bring 
forward  the  main  thermodynamic  conditions  at  the  sun,  under 
which  the  radiation  originates,  and  no  attempt  is  made  here  to 
interpret  them  fully. 

(8)  The  sun's  atmosphere  consists  of  a  gaseous  envelope 
containing  a  spherical  isothermal  shell,  at  the  mean  temperature 
of  about  7687°  C.  This  phenomenon  serves  to  explain  several 
of  the  common  solar  problems.  Above  the  isothermal  layer  the 
temperatures  decrease  rapidly  to  0°  C.,  and  below  it  the  tem- 
peratures increase  to  very  great  values.  Hence,  low- tempera- 
ture spectra  are  projected  upon  a  uniform  continuous  spectrum 
at  a  definite  temperature  of  7687°  for  a  background;  the  flash 
spectrum,  at  500  kilometers,  develops  for  the  light  gases  above 
the  heavy  metallic  strata  o.f  the  photosphere;  the  granulation 
is  an  optical  effect  of  segregation  within  the  isothermal  layer; 
the  sun  spots  have  two  independent  parts:  (l)  The  lower  on 
the  under  side  of  the  isothermal  layer,  generating  a  convectional 
vortex  like  the  terrestrial  hurricanes,  and  forming  the  dark 
umbra;  (2)  the  upper,  located  above  the  isothermal  layer,  and 
falling  into  the  umbral  vortex,  as  into  the  depression  at  the 
center  of  a  hurricane,  thus  producing  the  penumbra;  the  faculae, 
flocculi,  and  prominences,  are  convectional  effects  of  rising  gases 
originally  at  very  high  pressures  and  temperatures,  discharging 
through  the  isothermal  layer,  and  seeking  a  new  equilibrium  above 
it;  the  general  circulation  of  the  sun  must  conform  to  the  con- 
tinuous production  of  this  isothermal  layer  at  every  point  of 


THERMODYNAMIC   PROCESSES   IN   SOLAR  ATMOSPHERE  9 

the  photosphere;  the  radioactive  discharge  within  the  isothermal 
layer  sets  free  vast  numbers  of  electrons,  which  make  up  the 
surface  electric  charge  of  the  sun,  with  coronal  dispersion  to 
space  under  the  pressure  of  solar  radiation;  the  movements  of 
free  electrons  produce  the  Zeeman  effect  in  the  sun-spot  vortices ; 
they  cause  the  coronal  polar  magnetic  fields,  and  the  general 
polarization  throughout  the  sun. 

Historical  Remarks 

The  problems  of  the  physical  constitution  and  the  processes 
in  the  Sun  have  engaged  the  attention  of  scientists  for  three 
centuries.  Since  the  invention  of  the  telescope  the  statistical 
data  regarding  the  sun-spots,  and  the  solar  rotation  in  different 
latitudes,  have  been  studied ;  with  the  application  of  the  spectro- 
scope the  data  regarding  the  prominences,  or  hydrogen  flames 
on  the  disk,  have  been  classified,  since  1871;  with  the  develop- 
ment of  the  spectroheliograph  the  constitution  of  the  visible 
spots,  faculae  and  flocculi,  has  been  closely  analyzed,  and  the 
pressures  in  certain  levels  of  a  few  gases  have  been  approxi- 
mately determined;  with  the  discovery  of  the  ionization  of 
molecules,  and  the  existence  of  free  electrons,  together  with 
the  radioactive  processes  in  atoms  and  molecules,  and  the  struc- 
tural arrangement  of  the  positive  nucleus  with  the  circulating 
negative  electrons  in  the  atoms,  including  their  abrupt  redis- 
tribution by  the  association  of  atoms  into  molecules,  or  the 
dissociation  of  molecules  into  atoms,  under  certain  pressures, 
densities,  thermal  efficiencies,  the  physical  field  has  been  en- 
larged to  an  extraordinary  degree;  -the  attack  upon  the  thermal 
spectrum  by  means  of  the  bolometer,  and  the  measurement  of 
the  amount  of  radiation  received  at  stations  in  the  lower  atmo- 
sphere of  the  earth  by  the  pyrheliometer,  have  afforded  much 
information  regarding  the  coefficients  of  emission  and  absorption 
of  the  solar  radiant  energy  in  the  earth's  atmosphere;  some 
progress  has  been  made  in  regard  to  the  synchronism  of  the 
solar  variations  as  observed  and  the  corresponding  effects  upon 
the  meteorological  or  climatic  conditions  prevailing  as  tern- 


10  A   TREATISE   ON  THE   SUN?S  RADIATION 

peratures,  pressures,  vapor  pressures,  circulation,  electric  and 
magnetic  variations  in  different  terrestrial  latitudes  and 
longitudes. 

In  spite  of  the  extensive  literature  regarding  the  material 
just  mentioned,  it  must  be  admitted  that  the  scientific  knowl- 
edge available  is  really  superficial,  because  there  has  been  no 
thoroughly  comprehensive  analysis  of  the  many  physical  con- 
ditions which  must  exist  in  order  to  produce  the  very  complex 
phenomena  that  are  observed  on  the  sun.  All  the  above 
methods  depend  upon  visual  phenomena,  but  these  are  of  limited 
scope  in  respect  of  the  fundamental  conditions  in  the  solar 
gases  which  produce  them.  It  has  seemed  to  the  writer  that 
thermodynamics  constitutes  the  royal  road  between  the  earth 
and  the  sun,  and  that  the  system  of  equations  which  can  repro- 
duce the  general  conditions  in  the  earth's  atmosphere  should  be 
able  to  do  the  same  in  the  sun's  atmospheres.  With  this  purpose 
in  view,  the  terrestrial  dynamic  and  thermodynamic  relations 
have  been  studied,  in  order  to  develop  such  a  system  of  equa- 
tions as  is  effective  and  at  the  same  time  simple  enough  for 
practical  computations.  In  certain  papers  published  in  the 
U.  S.  Monthly  Weather  Review,  1905,  1906,  and  in*  Bulletins 
No.  3,  No.  4,  of  the  Argentine  Meteorological  Office,  1912,  1914, 
and  in  my  Meteorological  Treatise,  1915,  such  equations  and 
their  applications  have  been  explained  for  the  earth's  atmo- 
sphere. The  present  treatise  contains  their  modification  so  that 
they  become  equally  valid  in  the  sun's  envelope,  thereby  prov- 
ing that  this  thermodynamic  system  is  of  universal  application, 
as  to  any  star  where  the  surface  gravitation  is  known  from  the 
astronomical  masses  of  binary  stellar  systems.  While  the  labor 
of  developing  this  method  of  computation  has  been  very  con- 
siderable, it  has  now  become  relatively  simple  in  consequence  of 
the  discovery  of  certain  laws  which  regulate  the  distribution  of 
the  fundamental  quantities. 

Derivation  of  the  Non-adiabatic  System  of  Equations 

The  change  from  the  adiabatic  to  the  non-adiabatic  system  of 
equations  consists  in  making  the  gas  coefficient  of  thermal 


THERMODYNAMIC   PROCESSES   IN  SOLAR  ATMOSPHERE          11 

efficiency  R  a  variable  in  the  Boyle- Gay  Lussac  Law,  P  =  p  R  T. 
In  the  adiabatic  system  Ra  =  constant  for  each  gas,  and  it  is 
generally  appropriate  to  the  conditions  of  laboratory  practise,  but 
these  do  not  conform  to  those  prevailing  in  free  atmospheres.  If 
the  conditions  in  a  free  atmosphere  were  adiabatic,  then  the  tern- 

rr\  rr\ 

perature  gradients  would  be  at  the  fixed  rate, —    —  =  777-; 

#1  —  ZQ          Lpa 

where  Cpa  is  the  specific  heat  at  a  constant  pressure,  Cpa  = 
Ra  —  — ,  and  K  =  Cpa/Cva,  the  ratio  of  the  two  specific  heats. 
But,  as  a  matter  of  fact,  the  observed  temperature  gradients 

T*  T*  ft 

are    different,    such    that =  -Jh ,    so    that    Cp  = 

*i  -  ZQ        Cp1 

if 

R  -  — ,  and  R  must  be  a  variable  coefficient.      The   most 

compact  way  to  arrive  at  the  general  equation  of  condition, 
Treatise  (27),  is  as  follows: 

Take  all  the  terms  variable  in  Pv  =  R  T,  and  integrate 
between  two  levels  z\  .  26,  within  which  stratum  the  temperature 
gradient  remains  constant. 

(1)  Pdv  +  vdP  =  RdT+TdR, 

(2)  P10  Oi  -  w0)  +  *io  (Pi  -  Po)  =  -Rio  fa  -  To)  +  Tw  (Ri  -  jRo), 
We  have  the  term  for  the  external  work, 

(3)  P10(v1-vQ)  =  (Wl-WQ)=R10(Ta-T0)-v10(Pl-PQ),  (333), 
so  that  by  substitution, 

RIQ  (Ta  -  To)  =  £10  (Z\  -  To)  +  Tio  (Ri  -  Ro). 

The  equations  (2),  (3)  give  the  relations  between  the  work, 
the  hydrostatic  pressure  per  unit  density,  and  the  gas  efficiency 
in  the  stratum,  but  the  gravitation  acceleration  g  (21  —  z0)  does 
not  yet  appear  in  the  equation.  In  order  to  introduce  any  ex- 
ternal impressed  forces,  such  as  gravitation  or  circulation,  we 
proceed  as  follows,  by  adding  and  subtracting  Cpa  (Ta  —  TO)  = 
—  g  (zi  —  20).  Hence, 

(4)  OFx-PFo)  +  ^~  =  -  g(zi  -  86)  ~  (Cpa-Ru)  (Ta-T0). 

PlO 


12  A   TREATISE   ON  THE   SUN'S   RADIATION 

We  employ  the  following  forms   in   the  first  equation  of 
thermodynamics,  Treatise,  pp.  81,  82, 
(5)         (Gi  -  Co)     =  (JFi  -  Wo)  +  (U,  -  U0), 

(6)      (ft -ft)   =  (CP.- cp10)  (r.-  r.), 

(7)  (Wi  -  TFo)  =  (R*  -  Cpm)  (Ta-  To), 

(8)  (tf,  -  tf.)   =  (Cpa-  #,„)  (r.-  To)  =  Cvm  (T-a-  To), 
so  that  by  substitution, 

(9)  +  *(*-  so)  =  -  ^— ^  -  (W,  -  W0)  -  (U,  -  U0), 

PlO 

(10)  +  g&-  so)  =  -  ~^  -  (Ql  -  Co). 

PlO 

If  the  kinetic  energy  of  circulation  |  (q^  —  qQ2)  is  obtained  by 
observing  the  velocities  (qi  .  q0)  on  the  levels  (zt  .  ZQ)  respec- 
tively, a  secondary  value  of  —  (Qi  —  Qo)  =  —  (Qi  —  Qo)1  — 
2  (?i2  ~~  <?o2)  can  be  introduced,  so  that, 

(11)  g  (z  l-  So)  =  -  ^-P^°  -  *  (9l*  -  tf)  -  (<&  -  Q,Y, 

PlO 

and  this  is  the  equation  of  condition. 

In  the  terrestrial  atmosphere,  some  account  of  the  circulation 
can  be  obtained  from  high-level  observations.  In  the  sun's 
atmosphere,  while  circulation  has  been  somewhat  noted,  its 
conditions  as  to  level  are  so  uncertain  that  as  yet  no  attempt 
has  been  made  to  incorporate  it  into  this  Treatise,  although  it 
can  be  done  with  further  experience. 

We  may  note  that  the  following  potentials  have  been 
developed: 

V  for  g  (zi  —  2b),  the  external  potential  energy, 

K  for  \  m  (qi2—qo2),  the  external  kinetic  energy  of  circulation. 

J0  for  (Ui  —  Uo)a,  the  internal  potential  energy  of  atoms, 

Hm  for  (Ui—  Uo)m,  the  internal  kinetic  energy  of  molecules. 

It  appears,  therefore,  that  the  total  inner  energy  in  the 
atmosphere  (Ui  -  U0)  =  (Cpa  -  RIO)  (Ta  -  To),  sustains  the 
external  potential  energy  of  gravitation,  and  a  certain  thermal 
efficiency  Rw  (Ta  —  T0)  which  is  equal  to, 

(12)  +  ^^  +  (Wi  ~  Wo)  =  -  g  (z,  -  z»)  -  (U,  -  tf0)  = 

PlO 

Rio(Ta  —  TO), 


THERMODYNAMIC   PROCESSES    IN   SOLAR  ATMOSPHERE          13 

and  that  these  are  derived  from  the  state  of  the  inner  potential 
energy  of  the  atoms  Ja,  and  of  the  kinetic  energy  of  molecular 
translation  Hw.  Since  Ja  and  Hm  depend  in  the  earth's  at- 
mosphere upon  the  radiant  energy  from  the  sun  that  has  con- 
verged upon  the  unit  volume  per  unit  time,  it  is  proper  to  com- 
pute the  terrestrial  thermodynamic  data,  and  proceed  thence  to 
the  radiation  data.  Similarly,  the  thermodynamics  of  the  solar 
atmosphere  must  afford  the  corresponding  data  for  the  emission 
of  radiation,  which,  reduced  to  the  distance  of  the  earth,  should 
be  equivalent  to  the  terrestrial  thermodynamic  data. 

In  the  first  Treatise  on  "Atmospheric  Circulation  and  Radia- 
tion" the  terrestrial  data  have  been  summarized;  in  this  Treatise 
on  "  Solar  Radiation"  the  corresponding  solar  data  are  briefly 
given;  there  still  remain  certain  problems  of  the  electromagnetic 
radiation  and  its  relations  to  the  electrons,  ions,  atoms,  and 
molecules,  which  will  require  more  extended,  careful  examina- 
tion. The  formulas  given  above  can  be  illustrated  and  verified 
from  the  Tables  13-27,  Bulletin  No.  4,  0.  M.  A.,  1914. 

Preliminary  Summary  of  Conditions 

Fig  1  contains  a  summary  .of  the  results  computed  from  the 
balloon  ascension,  Uccle,  September  13,  1911.  The  data  are 
reduced  to  one  arbitrary  scale  by  the  series  of  factors  indicated, 
and  the  lines  show  the  course  of  development  from  the  sea  level 
to  the  vanishing  plane.  In  the  thermodynamic  section  it  may 
be  noted  that  the  free  heat  (Qi  —  Q0)  has  two  equivalent  areas, 
B  EC  =  B  AC.  The  abscissas  give  the  relative  values  of  each 
term  at  the  several  levels,  so  that  (Q\  —  Q0)  =  0  at  the  sea  level, 
and  it  is  g  (z=  —  z0)  on  the  vanishing  plane.  The  area  A  C  D  = 
RIO  (Ta  —  TO),  and  it  is,  also,  equal  to  the  area  A  B  D,  as  given 
by  formula  (12).  It  will  be  shown  how  the  pyrheliometer  at 
sea  level  measures  a  quantity  proportional  to  E  F,  while  the 
bolometer  measures  another  quantity  proportional  to  E  A; 
on  the  vanishing  plane  the  pyrheliometer  would  be  propor- 
tional to  B  G  and  the  bolometer  to  B  C.  We  shall  be  able  to 
indicate  the  physical  significance  of  the  line  G  F,  and  the  line 


14  A   TREATISE   ON  THE   SUN'S  RADIATION 

SECTION  I.    DYNAMIC  DATA  FOR  P  =  PRT,  (M.  K.  S.) 


Scale  for  P  -=-  10000 
»    K  -r-  10000 

„   „  p  -±_   10 

,.   ..  R  4- 

..  T  -f-   25 


80000 
75000 
70000 
65000 
60000 
55000 
50000 
45000 
40000 
35000 


9    10   11    12 


SECTION  II.    THERMODYNAMIC  DATA. 


80000  B 


Scale  (Q!-QO)  -v-1000 
«      (Wi-Wo)-r-lOOO 

1000 


5          6 


10     X  1000      Table  19 


SECTION  III.    KINETIC  THEORY  DATA. 


10        11        12        13 


FIG.  1.     Summary  of  the  Data  for  the  Atmosphere. 
Balloon  Ascension,  Uccle,  September  13,  1911. 


THERMODYNAMIC   PROCESSES   IN   SOLAR  ATMOSPHERE          15 

C  A,  together  with  their  relation  to  the  incoming  solar  radia- 
tion B  C. 

Summary  of  the  Data  on  Section  II 
Gravity  acceleration,  g  (zi  —  z0) .  . .  =  D  E  B  C, 

T)  7> 

Hydrostatic  pressure, —    — . .   =  D  E  B, 

Pio 

Work  of  expansion,  (Wi  -  JFo) .  . .  =  A  B  E  =  A  C  E, 

Total  inner  energy,  (Ui  -  U0) =  A  E  B  C, 

Free  heat,  (ft  -  Q0) =  EEC  =  ABC  =  DEC, 

Gas  efficiency,  R10(Ta-T0) =  A  C  D  =  A  B  D, 

Pyrheliometer  radiation,  equivalent 

to EFGB, 

Bolometer  radiation,  equivalent  to       E  A  C  B, 

Kinetic  energy  of  the  Poynting  flux  =  E  F  G  B, 
Potential  energy  of  the  Poynting 

flux =  FACG, 

Dissipated  energy  of  the  Poynting 

flux =  ADC, 

Kinetic  energy  of  solar  radiation . .  =  B  G, 

Potential  energy  of  solar  radiation  =  G  C. 

By  keeping  these  conditions  in  mind  the  reader  will  more 
readily  follow  the  argument  of  the  following  pages. 

The  Working  Equations 

There  are  only  two  important  steps  which  are  required  in 
order  to  transform  the  adiabatic  system  of  equations  into  the 
non-adiabatic  system  actually  applicable  at  the  sun.  The  first 
of  these  is  the  change  of  Ra  —  constant  into  R  =  variable  in  the 
Boyle-Gay  Lussac  Law,  P  =  p  R  T;  and  the  second  is  the 
multiplication  of  all  the  terms  in  Pv  =  R  T  by  7  =  Go/ go,  the 
ratio  of  the  acceleration  of  gravitation  at  the  sun  relatively  to 
that  at  the  earth,  so  that, 

p 

"  T    '  T 
If  Ra  =  constant  the  result  is  that  the  temperature  gradient 


16  A   TREATISE   ON  THE   SUN'S   RADIATION 

.    Ta  —  To  go  Ta  —  TO  Go 

is =  — -prr  on  the  earth,  and  = 77—  On 

Zi  —  ZQ  .         Lpa  Zi  —  ZQ  Lpa 

the  sun,  these  being  fixed  gradients  for  each  gas.  They  exist 
in  the  lower  levels  of  the  earth's  atmosphere  in  the  tropics,  and 
in  certain  strata  below  the  photosphere  of  the  sun,  but  above 
these  strata  there  are  in  every  atmosphere  isothermal  strata,  and 
above  these  non-adiabatic  strata,  where  no  such  temperature 
gradients  can  occur.  For  this  reason  the  adiabatic  formulas, 
which  have  been  universally  employed,  that  is,  wherever  Ra 
has  been  treated  as  a  constant,  are  inapplicable  in  two  deep 
strata  of  every  atmosphere.  In  order  to  pass  from  the  adiabatic 
to  the  non-adiabatic  conditions,  it  is  only  necessary  to  put  the 

Ta  —  To       adiabatic  gradient    ... 

factor  n  =  ^ TFT  =  — r 1 T- — r>  lnto  the  coefficients 

TI  —  TO       observed  gradient 

of  the  formulas. 

Adiabatic  and  Non-adiabatic 

(14)  Pressure.       log  P,  -  log  P0  =  ^-j-  (log  Z\  -  log  To) ; 

n  K 


log  P,  -  log  Po  =  ^  (log  Ti  -  log  To). 


(15)  Density.        log  pi  —  log  p0  =  ~T    (log  TI  —  log 


n 


log  Pl  -  log  po  = (log  Ti  -  log  TO). 

K  J. 

(16)  Thermal  Efficiency.        log  7?i  -  log  ^0  =  0; 

log  R,  -  log^o  =(»-!)  (log  7\-  log  To). 

The  importance  of  this  step  is  seen  in  the  fact  that  the 
entire  thermodynamic  and  radiation  systems,  instead  of  depend- 
ing upon  a  set  of  universal  constants,  become  involved  in  a 
series  of  variable  coefficients.  The  constants  apply  only  within 
the  adiabatic  strata,  while  the  variable  coefficients  are  neces- 
sary in  every  non-adiabatic  stratum,  that  is,  wherever  the  natural 
temperature  gradients  differ  from  the  fixed  adiabatic  rate.  It 
is  this  fact,  namely,  the  adoption  of  the  usual  laboratory  adia- 
batic equations  for  free  atmospheres,  that  has  prevented  meteor- 
ology from  entering  the  region  of  the  exact  sciences.  Change 


THERMODYNAMIC   PROCESSES   IN   SOLAR  ATMOSPHERE          17 

Ra  from  a  constant  into  a  variable,  and  the  physics  of  atmo- 
spheres becomes  a  perfectly  simple  branch  of  thermodynamics. 
Examining  the  dimensions  of  the  terms  in  the  Boyle-  Gay 
Lussac  Law,  we  have  for  the  absolute  temperature, 

P         M       Ls     T2 

T  =     =    *''*  degree  =  l  degree> 


so  that  the  absolute  temperature  in  degrees  has  no  dimensions, 
being  only  the  ratio  between  the  pressure  and  the  product  of 
the  density  with  the  thermal  efficiency.  The  last  is  merely  the 
velocity  of  the  molecules  per  degree,  which  pertain  to  the  num- 
ber in  the  mass  p  when  producing  a  pressure  P.  It  is  evident 
that  in  the  laboratory  conditions,  the  gas  enclosed  in  a  vessel 
of  impermeable  walls  is  very  different  from  the  same  gas  in  the 
lower  strata  of  a  free  atmosphere,  which  absorbs  or  emits  radiant 
energy  to  space.  As  a  matter  of  fact,  the  R  begins  at  full  adia- 
batic  value  in  some  low-lying  strata,  and  thence  diminishes  to  0 
on  the  vanishing  plane  of  the  gas.  If  R  is  variable,  so  are  all 
the  following  quantities  also  variables. 

(18)     Specific  heats,  Cp  =  -£-:  R}  Cv  =   -       -  R. 

K  JL  K  JL 

Thermal  efficiency,  K  =  m  R,  where    m  =  the  molecular  weight. 

•& 

Number  of  molecules  per  unit  mass,     N  =  m  v  n  =  -r. 

Number  of  molecules  per  unit  volume,  n  =  N/m  v  =  -^-j,  ~  ]T~- 
Boltzmann's  entropy  coefficient,  k  =  K/N. 


where 


Planck's  Wirkungsquan turn,  h  =  — (  —       -J    , 

c  ^     a     ' 

c  =  the  velocity  of  light,  a  =  1.0823  (Planck), 

a  =  variable. 
Wien-Planck  coefficients,  Ci  =  8  TT  c  h,  c2  =  c  h/k. 

6  a.  c\ 
Stefan  coefficients,  a  =      -  -, 

a  =  —  in  JQ  =   <r  T4. 


18  A  TREATISE   ON  THE   SUN'S  RADIATION 

The  thermodynamic  equation  for  the  conservation  of  energy, 

(19)          TdS  =  dQ  =  dW  +  dU  =  PdV  +  dU, 
is  limited  by  Planck  as  follows,  in  certain  analyses. 

Adiabatic  conditions:  dQ  =  0  =  PdV-\-dU,  so  that  the 
work  of  expansion  exactly  compensates  for  the  change  in  the 
inner  energy,  and  this  is  correct  in  the  lower  adiabatic  strata. 

The  entropy  equation,  developed  f  or  d  V  =  0,  so  that  T  d  S  = 
d  Q  =  d  U,  and  the  free  heat  is  equal  to  the  inner  energy.  This 
is  true  outside  the  vanishing  plane  of  the  gaseous  atmosphere, 
for  the  radiation  in  the  aether,  but  it  is  not  correct  within  the 
non-adiabatic  regions  where  d  W  =  P  d  V  has  real  values.  The 
two  fundamental  developments  follow: 

Bigelow  Planck 


-     -  4 
dV=T"TdV  WF/£7~r        V     ~  V 

*     kN 


dU      TdU      T          '\dU'y      T       2(Z71-£70) 

3         K  _K__ 

2  (U,  -  f/o)  ~  P  V 


It  is  evident  that  within  the  non-adiabatic  strata,  including 
the  isothermal  layer,  it  is  not  allowable  to  suppress  first  one 
term  and  then  another,  as  d  Q  and  d  F.  Hence,  the  following 
non-adiabatic  monatomic  thermodynamic  formulas  must  prevail: 

(26)     Pressure.    P  =  p  RT  =  knT  =  k^  =  ^  =  ^  K  T 


=  RTmnmH  =  KT^=  ~Y^P'   LTTU 
(27)     Kinetic  Energy.   P  V  =  K  T  =  mR  T  =  kn  VT  =  kNT 


THERMODYNAMIC  PROCESSES  IN  SOLAR  ATMOSPHERE          19 

(28)  Inner  Energy.    (U,  -  U*)  =  H  V  =  \  P  V  =  £-  U  = 

4  Z  p 

^KT--jPm9-  jkNT  =  C,mT  =  C,pTV.  [^ 

(29)  Specific  Heat.   C,  =  ^=/-°  =  ^=£°  =  -|  ^|  = 

tfYi  J.  P  J-   v          £  m  J. 

3  K       3       _3P         3  k  N        H_     r     L2    -i 
~2m='~2         '~2'pf='~2    m     =  ~pf'     L^deg.J' 

(30)  Specific  Heat.  Cj  -  £*  "  T'«  "  4  *  S'  C#-C»  =  ^. 

Z  &    Til  £i         Til 


(31)  Volume.    F  =  m,  =  ^  =  |    [g]. 

Energy,    i/  =  Fw  =  Fa  T<  =  3  p  V 

(32)  Work.   W  =  pdV.  [^ 

(33)  Entropy.  5=F5  =       a 


All  these  formulas  are  verified  throughout  the  series  of  non- 
adiabatic  computations  included  in  this  Treatise. 

The  Initial  Constants  and  Coefficients  for  Solar  Atmospheric 
Computations 

In  order  to  prepare  the  necessary  constants  and  coefficients 
which  are  preliminary  to  the  computations  implied  in  the 
preceding  formulas,  and  applicable  in  the  gaseous  atmospheres 
of  the  chemical  elements  on  the  sun,  the  constants  which  are 
accepted  as  standard  under  the  earth's  gravitation  were  trans- 
ferred to  the  sun's  gravitation  by  the  factor  7  =  G/g0  =  274.S43/ 
9.806  =  28.028,  as  indicated  in  the  four  sections  of  Tables  1,  2. 
The  formulas  are  given  for  the  several  quantities,  and  these 
begin  with  P  =  101323.5  kilograms  per  square  meter  on  the 


20  A   TREATISE   ON   THE   SUN'S   RADIATION 

earth,  which  is  one  atmosphere  under  standard  conditions; 
multiply  by  7  and  P  =  2839900  kilograms  per  square  meter  on 
some  solar  level  of  equilibrium,  not  necessarily  the  plane  of  the 
photosphere,  where  P  =  6  atmospheres.  The  level  of  28.028 
atmospheres  occurs  at  very  different  distances  below  that  plane, 
namely,  —  5000  kilometers  below  for  hydrogen  (diatomic),— 
2500  below  for  helium,  —  420  for  carbon  (assumed  monatomic), 
—  250  for  calcium,  —  150  for  zinc,  —  90  for  cadmium,  and  —  50 
for  mercury.  These  differences  all  depend  upon  the  atomic 
and  the  molecular  weights.  In  order  to  avoid,  as  far  as  it  is  pos- 
sible to  do  so,  the  complicated  questions  regarding  the  values  of 
the  specific  heats  Cp,  Cv,  and  their  ratio  K  =  Cp/Cv  at  solar 
temperatures,  the  monatomic  elements  were  selected  for  these 
computations,  wherein  K  =  1.666+  under  all  temperatures, 
since  the  atoms  are  also  molecules.  There  are  two  exceptions 
in  this  selection.  It  was  found  necessary  to  treat  hydrogen  as 
monatomic  HI  below  a  certain  level,  but  as  diatomic  HZ  above  that 
level,  as  will  be  fully  explained  in  connection  with  the  subject 
of  the  origin  of  the  solar  radiation.  For  convenience,  carbon  at 
m  =  12.00  was  taken  as  if  it  were  a  monatomic  element,  not 
only  below  its  plane  of  transition  but  above  it  as  well.  It  is 
probably  monatomic  in  the  adiabatic  levels,  but  not  mona- 
tomic above  its  reference  plane.  However,  it  is  difficult  to 
assign  its  molecular  composition  and  its  specific  heats  at  the 
high  solar  temperatures,  but  it  fills  a  rather  wide  gap  in  the 
series  of  curves  to  take  carbon  as  if  it  were  a  monatomic  element 
at  m  —  12.00.  As  our  purpose  is  to  discover  the  primary  laws 
of  the  distribution  of  the  chemical  elements  near  the  surface  of 
the  sun,  there  is  no  reason  why  a  monatomic  element  should 
not  be  assumed  at  m  =  12.00.  If  the  actual  heavy  molecule 
of  C  can  be  assigned,  and  its  vanishing  height  above  the  photo- 
sphere established,  that  part  of  the  computation  can  be  ex- 
tended to  meet  these  general  facts.  Similar  remarks  are  appli- 
cable to  any  other  chemical  element  on  the  sun. 

It  will  be  seen  from  Table  1,  section  I,  that  the  factor  7  = 
28.028  serves  to  transform  the  following  constants  from  ter- 
restrial to  solar  values:  pressure  P,  molecular  volume  V,  thermal 


THERMODYNAMIC  PROCESSES  IN   SOLAR  ATMOSPHERE          21 

efficiency  K,  temperature  J,  molecular  kinetic  energy  per  degree 
e  =  EQ/T,  Boltzmann's  entropy  constant  k,  translational  kinetic 
energy  of  the  molecules  #,  total  inner  energy  J7,  gravity  ac- 
celeration G;  the  factor  72  is  required  for  the  kinetic  energy 
P  V  =  K  T,  the  kinetic  energy  of  one  molecule  £0;  the  factor 
7*1  is  for  the  number  of  molecules  per  unit  volume  w;  the  factor 
yla  for  the  Wien-Planck  constants  in  the  energy  spectrum  c\,  c%\ 
the  factor  y'3  for  Planck's  Wirkungsquantum  ;  and  the  following 
require  no  factor  whatever,  mechanical  equivalent  of  heat  A,  the 
specific  heat  ratios  K,  K/(K  —  1),  I/(K  —  1),  the  number  of  mole- 

cules per  unit  mass  N,  the  molecular  weight  of  hydrogen  mH=j^, 

and  the  velocity  of  light  c.  In  adiabatic  diatomic  conditions 
for  hydrogen  the  values  of  the  specific  heat  ratios  are  given  in 

rji    _  rri  ri 

section    II.     The    adiabatic    gradient    —  —    -  =  —  77-  is  re- 

Zi  —  zQ  Cp 

corded  in  section  III  for  eight  gases,  and  they  are  utilized  in  the 
respective  computations. 

Section  IV  contains  the  important  thermal  coefficients  for 
the  same  list  of  gases,  which  are  dependent  upon  the  molecular 
weight,  and  are  therefore  variables  from  one  gas  to  another. 

These  are  the  density  p  =  m  n  m^,  the  volume  v  =  —  ,  the 


thermal  efficiency  R  =  —  ,  the  specific  heat  at  constant  pressure 

K  1 

Cp  =  -  -  R,  and  at  constant  volume  Cv  =  —  —  R.     The 
K  —  1  K  —  1 

factor  7  is  required  for  v,  R,  Cp,  Cv,  and  7"*  for  p.  The  specific 
values  are  taken  as  they  are  more  convenient  in  the  computations. 
The  chemical  elements  were  chosen  with  m  ranging  from  HI  = 
1.00  to  Eg  =  198.41,  sufficient  in  number  to  bring  out  clearly 
the  relations  which  prevail  in  the  families  of  curves  developed 
with  the  parameter  m,  all  the  other  elements  being  easily  inter- 
polated into  their  proper  places.  These  elementary  constants 
and  coefficients  apply  to  certain  non-adiabatic  conditions  of 
equilibrium,  and  they  locate  one  point  for  each  term  of  the  gas, 
dynamic  or  thermodynamic,  at  some  unknown  level  in  the  solar 
atmosphere.  There  is  such  a  point  of  equilibrium  fulfilling  the 


22 


A  TREATISE   ON   THE   SUN'S   RADIATION 


5 


0 


:   : 


iO       »O  O  T-H  ^  to 


<M  C5 

<N 


l>       CO       OO  <N  CO  l>  <N  CO  *H         1>  O5 

<M*       1C       CO  Ci  CO  I-H  O  O  O          CD  CO 

d          (N 


(N      l>  1-1 


i— i  I-H       CD       COC^i— ttOto^Oi 

O  CO       O5       TjH  CO  CO  W  W  CO  i-J 

tO  TH      CO      (NCOCOOOOO 


(M 


s  s 

x  2 

CO  (N 

^3  Is* 


i— I   C<l   TH  CO 


J-       <- 


PQ 


^E^ 


, 


*»      ^ 


7 
" 


"rt 

OJ 

1    c 
"c3 

" 


j 


if  Is    ==    s 

ijE    I.S      .S 


rj< 

O 


CQ 


THERMODYNAMIC   PROCESSES   IN   SOLAR  ATMOSPHERE          23 


oobob 


§00  i— i  OO  OS  CS  C^ 

CO  00  CO  CO  CO  rH 

Is—          CO  CO  03  CO  CO 

<H       ei  <N  i>  cd  co  06  co 

CO          C^  <N                         '"' 

I              I  I  I 


X       X 

»O          <N 


»o       co     os 

00  CO         rH 

CO  ^ 


CO  00  Is* 


?     2 


*•>• 


8  II 

s^ 


00 


I; 


OO 


odd 


10 

»o 


CO  CO 
i-J  CO  (N 


odd 


*O  »O 


1     1 


I 


O 


8    8    8 

o  o  o 


irH 


58S8 

^  CO  (M  iO 

O  00  O  CO 


CO  CD  i^  ^O  OS  CD  CO  ^^ 
i— i  CO  1O  OS 

I   I   I   I   I   I-  I   I 


6 


8    S    8 

OS  OS  »O 


f^co0^  Si 
cqoq<NJ>W'^^Os 

r-5i-5ddi-5i— 5i-5i-5 
I     I 


»OOSi— i^Ot^-COiOO 
iOCO£JCOi— ^Osl>Cl 
O  C^  CO  C^  *O 

ddrHioosdco'1* 

rH   CO  >O  OS 
I        I        I        I        I        I        I        I 


I  I 

rt  cd 

§  § 

8.a|^| 

5  *3  O  33  O 

£   o  rt  £  ca 

3   -M  C    3    C 

«?  .?3  o  a  o 


c  c  ^r"i, 


^1^3131 


&3 

CJ 

z 

^ 

£ 

5 

a 

3 

'fl 

I 

H 

fc 

w 

1 

3 

B 

1 

4J 
§ 

A   TREATISE   ON   THE   SUN'S   RADIATION 

£:    = 


8 


00  CO  i-<  i— i  CO 
O2  CO  to 

O  CO  00  O  O4 

dOOTHr-IrH^IN 


OOOOOOOto^? 


CO  CO  00  CO  i—  1  rH  CO 

co  co  *"H  co  os  co  to 

r-(C^O5(M'-Hl^i> 

CO  CO  t^»  CO  fH  ^^  O^ 

COCOOCOOOOd 


OOOOOOOiO^ 


Assumed  monatom 
Diatomic 
Monatomic 
Assumed  monatom 
Monatomic 


8^5     ^^     T-H     CO     ^^     t>»     T~^ 
O    O   O    O   rH    rH   CO 

ddddcoddd 


1   1   1   1 


Dia 
Mo 
Assu 
Mo 


C     C 


<M 


i 

-js  •£  ^  y 


THERMODYNAMIC   PROCESSES   IN   SOLAR  ATMOSPHERE          25 


O^  "^^  O^  00  T}^  O5 

T-H   1>    rH   O5  l>;   TjJ 

(N   rH    rH   O   O   O 


00  00 
<N  CO  00  CO  T-H 

00  TJH  I>  Oi  iO 

erH   1C   rH    lO   Oi 
r- 1   lO   *O   rH 
CO  rH 


rH   00   *0   CO    »0   O 

t>  CO  t>-  O 


>O  00    _    _-    _- 
t^  00  OS  »O  CO 

i— I    LO    T-H    IQ    C^ 


s 


C<     rH 


rH    IO   rH    O   O  O   O 


a 

I 

1 


.y  £ 


.a 

o 

i 

ago 

«1 


^2  g  5  g 

en    aj    o    W3    O 

Islls 


COCOOOOOIO»OO 
CO  CO  CO  ^H  CO  ^^  C^  00 
OtO-^t^-^r^OSGO 

i—  IT—  IT—  ICOr—  tOO'—  ' 

O5COCOOOCOT-IOOO 


fNcDOO 

i—  i  O  CD 

TJH    <M 


I>  CO  rH 


ca 

§ 

.a  g. 


Isi  II 


.  a 


26 


A   TREATISE   ON  THE   SUN  S  RADIATION 


0» 

HO 

U 

3 

I 

2 

W 

Number 

W 

3M 

| 

a 

i 

c 

3 

1 

to 

4» 

s- 


1 


OS  OS  CO  ^  ICOCOOOCO 

oo  '"^  oo  i>»  oo  os  *o  co 

CO  O  »O  OO  iO  Tt<  i—  ICO 

!>•  CO   i-H  *O   i—  1   OS  !>•  ^ 


t^O 
l>-  O 

IO    TjH 


co 
C<J  l>  O  O  OS 

00  !>•  <N  00  <N  CO  <N 

^  00  CO  ^  rH  GO  rH 

*^  o^  ^^  o^ 


CO 
CO  OS  i—  i 


rHi—  IOOO 


*Q  VH 


ssumed  monatom 
iatomic 
atomic 
med  monatom 


onatomic 
u 

tt 


A 

D 
M 
A 
M 


C    C 
flj     flJ 


' 


(N  00  1> 


t^OSTt^TtHOS 


l>  CO  O  00  »O  1C 

coi>ppcqi>osco 

f>»  IO  OS  CO  CO  OS  i~^  CO 

CO   T— I   O   O   CO   i-H   »H 

<M  O  CO  ^H 


umed  m 
iatomic 
onatomic 


mo 


to 
ii 


Ass 


D 
M 
A 
M 


THERMODYNAMIC  PROCESSES  IN  SOLAR  ATMOSPHERE         27 

Boyle-Gay  Lussac  Law,  y  (P  =  P  RT),  and  from  this  point, 
by  the  method  of  trials,  it  has  been  found  to  be  practical  to  proceed 
upwards  along  certain  curves  to  the  vanishing  plane,  or  down- 
wards to  considerable  depths,  where  this  gaseous  law  begins 
to  give  way  to  the  liquid  and  viscous  conditions  at  great  pres- 
sures, densities,  thermal  efficiencies,  and  temperatures.  The 
following  chapter  presents  such  computed  values  for  the  thermo- 
dynamic  and  radiation  data  as  are  indicated  in  the  formulas. 
In  all  the  adiabatic  levels  there  are  certain  constant  values,  but 
generally  the  solar  atmospheres  are  in  equilibrium  only  by  means 
of  a  series  of  variable  coefficients  which  will  be  quite  fully  ex- 
amined. It  may  be  stated  that  the  computed  results  are  in  ad- 
mirable agreement  with  the  data  obtained  by  means  of  observations 
made  with  the  spectroheliograph,  but  that  they  far  outrun  them  in 
efficiency,  as  to  a  multitude  of  details  which  are  of  great  value  in 
all  questions  of  solar  physics,  regarding  the  origin  of  the  circulation, 
radiation,  and  variations  in  the  visual  and  the  thermal  spectra. 

The  most  convenient  formulas  for  computing  the  terms  in 
the  first  law  of  thermodynamics  are  as  follows  : 

(34)  Free  heat  (ft  -  Q0)  =  (Cpa  -  Cp1Q)  (Ta-  T0)  = 

(Si  —  So)  7*10, 

(35)  Work  (W,  -  Wo)  =  (R»  -  Q>w)  (r.  -  T0)  - 

Pvs  (t>i  —  »o), 

(36)  Inner  energy     (U,  -  Uo)  =  (Cp,  -  RIO)    (Ta  -  T0)  = 

(ft  -  ft)  -  (Wi  ~  Wo), 
and  for  the  second  law  of  thermodynamics, 

(37)  Entropy  (S,  -  So)  =  &-=-&  =  (Cpa  -  Q10)  (T°  ~  T°]  . 

J-  10  V         -t  10         ' 

From  these  the  potential  of  radiation  is 


/OON  K  o  o       p        rT* 

(38;  AIO  =  -  =  --  /io  =  c  L  . 

V  1—  VQ  Vi  —  V0 

The  coefficient  c  and  exponent  a  are  computed  by  the  methods 
of  Bigelow's  Meteorological  Treatise. 
From  the  trial  equation 

(39)  log  C  =  log  K1Q  -  A  log  r10, 


28 


A  TREATISE  ON  THE  SUN'S  RADIATION 


we  have,  by  using  successive  values  of  KiQ  and  T10, 

log  (log  K!-  log  K0) 
lo*A    =  log  (log  Ti  -  log  To)' 

which  gives  certain  values  of  A  that  can  be  combined  with  the 
corresponding  log  C  in  Equation  (39).  These  values  of  (-4, 
log  C)  are  plotted  on  a  diagram,  and  the  mean  line  is  drawn,  from 
which  a  table  for  (a,  log  c)  is  made.  Trial  values  of  «i  are 
assumed,  and  log  c  is  computed  from 

(41)  log  c  =  log  K10  —  a  log  TIQ. 

In  this  table  the  corresponding  value  of  a  is  interpolated, 
and  if  «i  =  a  the  trial  value  is  correct.  Two  trial  values  («i,  a^) 
are  generally  enough  to  produce  the  pair  values  (a,  log  c)  on  the 
same  horizontal  line  of  the  table  of  the  gas.  Examples  of  the 
results  of  these  computations  will  be  given,  and  they  are  un- 
usually interesting  in  determining  the  source  of  the  solar  radiation 
in  the  different  gases  and  their  relation  to  a  in  the  Stefan  Law, 

(42)  /o  =  a  T*. 


Notation  and  Fundan 

(43)  Field  strength  . 
Impressed 
forces  

Cental  Formulas  for  the 
Electric  Field 
EI      (Ex.  Ey.  Ez)i 

e0 
e 

E  =  D-  = 

K 

Ei  +  eo  +  e 

^-  =  b(bx.by.bz) 

ot 
47rC  =  4-ir.cE 
4?r  pv 
**j 

47T/0   =   D  +  4lT 

(C+pv+j] 

Electromagnetic  Field 

Magnetic  Field 

Hi                    (Hx.HZ.Hy)l 

ho 
h 

H=*-  = 

tf 

Hi  +  //o  +  h 

-—  —  B  (Bx.  By.  Bz) 

oi 

Motion   of   the 
medium  
Total  force.... 

(44)  Displacement 
current  
Conduction 
current  

Convection  cur- 
rent   

Motion   of   the 
medium  
Total  current.  . 

47T  g 

47T<70  =  B  +  iirg 

THERMODYNAMIC  PROCESSES  IN  SOLAR  ATMOSPHERE         29 

The  impressed  forces  and  the  motion  of  the  medium  will 
not  be  required  in  the  problems  concerning  the  radiation. 

(45)  Maxwell's  Laws  of  Circuitation 

I     47r/o  =  £>  +  47T  (C  +  PV+J)  =  curl  (H  -  fc) 
47r/   =  £)  +  47r(c_|_  pv)         =  curl#1 

47rr1  =  Z)-f47rC  =  curl  H 

(46)  II    47rG0  =  B  +  47rg  =  -  curl  (E  -  *>) 

47rG  =  B  =  -  curl£t 

47rr2  =  B  =  -  curlE 

(47)  Potential  Energy 


J  =       .(ExDx  +  EyDy  +  EZDZ)  d 


6r  Sir 

(48)     Kinetic  Energy 

K  =  ^  fff(H*  Bx  +  H,B,  +  Hz  B,)d  r  = 


(49)  Joule's  Heat 

Q  =      fff  (Ex  Cx  +  EyCy  +  EZCZ)  d 

i£ 

(50)  Activity 


(51)  Scalar  product 

(A  B)  =  A  B  cos  A  B  =  Ax  Bx  +  Ay  By  +  Ax  Bz. 

(52)  Vector  product 

[A  B]  =  (AyBz  -  AzBy)  +  (AZBX  -  Ax  Bz)  + 

(AxBy-  AyBx)  =  V.AB. 


(53)  Divergence  div  A       =        *  +          +         . 

(54)  Curl-rotation  curl  4,  =          +         ;  curl  ^y  =          " 


30  A   TREATISE   ON   THE   SUN'S   RADIATION 

The  five  fundamental  equations  are 

Usual  Units  Lorentz*  Units 

(55)  1.  Divergence     div  D  =  p  div  D  =  p 

(56)  2.  div  B  =  o  div  B  =  o 

(57)  3.  Curl  curl  D=  -H  curl  D  =  -  ~  H 

c 

(58)4.    "  curl#  =  £>-t-47rpz;     curlF  =  i  (Z> 

£~ 

(59)  5.  M£rhcaenical  /?  =  E  +  [v.H]  F  =  D  +  i  fo.ff] 

Poynting's  Equation  for  the  Flux  of  Radiation 
The  following  demonstration  of  the  Poynting  equation  is 
according  to  Richardson's  exposition,  Electron  Theory  of  Matter, 
page  202. 

This  theorem  proves  that  the  rate  at  which  work  is  done  on 
the  electric  charges  within  a  given  volume  d  r  has  two  com- 
ponents, first,  the  rate  at  which  energy  flows  into  the  volume 
d  T  through  the  surface,  and  second,  the  rate  of  loss  of  the 
energy  by  the  electromagnetic  field  within  the  volume.  Take 
the  mechanical  force  acting  upon  p  electrons  in  d  r.  s 

(60)  Fp  =  p(E  +  [v.ff])dr, 

E  and  H  represent  the  total  electric  and  magnetic  forces. 

The  activity,  or  rate  of  working  of  the  forces  in  the  volume,  is, 

(61)  A  =  ///  vp(E+[c.H])d  r. 

c  =  the  velocity  of  propagation  in  the  medium. 

The  second  term  is  always  zero,  so  that  the  activity  is 


(62)     A  - 

Expand  by  the  formulas  for  curl, 


THERMODYNAMIC  PROCESSES  IN  SOLAR  ATMOSPHERE         31 

Collect  the  terms  and  integrate  by  parts,  rearranging  them, 

(64)  A  =  -^ff(EyHz-EzHy)dydz+(EzHx- 

ExHz)dzdx  +  (Ex  Hy-  EyHjdxdy 


Substitute  dydz  =  ldS,    dzdx  =  mdS,    dxdy  =  ndS, 
surface  integral. 

From  H*  =  Hx*  +  Hy*  +  H,*,  we  hzve2H~  =  2 


Ql  O   * 


Since  rot  E  = ,  we  can  employ  the  components;   sub- 

8  * 

stitute  them, 

i3-±-ffrtfaM.+ 

IrJ  J  J    (  \  Qt 


U--[dr. 


A  =  r,  ff  w»ds-r*  ///9T(*  ff2  +  *  £2)  dT- 

Change  the  general  E  .  H  to  the  specific  K  E0,  M  HQ. 


-L  V.EH-H-j. 


E  and  H  are  perpendicular  to  each  other,  and  the  propaga- 
tion is  perpendicular  to  both.     The  internal  energies  are  equal, 


32  A  TREATISE   ON  THE   SUN?S  RADIATION 

In  a  beam  of  parallel  plane  polarized  light  the  electric  and 
magnetic  vectors  move  in  equal  periodic  waves, 

E  =  Eo  cos  (<at  —  x),H  =  H0  cos  («  /  -  x). 
The  resultant  flow  of  energy  perpendicular  to  E  H  is 
E2  =  £02  cos2  («/-*)  =  #02  cos2  («/-*)  =  #2. 
Its  average  value  over  a  long  interval  of  time  is 

(69)         c  E2  =  c  E02-f  cos2  (co  /  -  x)  d  t  =  c.  |  E02  =  c  \  //02. 
The  energy  flows  along  with  the  velocity  c,  as  that  of  light. 

Pressure  of  Radiation 

If  E  —  the  average  value  and  E0  the  maximum  value,  it  is 
proved  that  the  pressure  due  to  radiation  is 


(70)  p  =~E2  d  t  =  -j^o*  cos2  co  t  d  t  =  |  £02. 

The  energy  per  unit  volume  is 

K  u  E2       H2 

(71)  u---Et  +  £-Hf-  !-*=-. 

Sir  Sir  4  TT        4?r 

Flux  of  light  transported  on  the  average  over  the  unit  area 
in  the  unit  of  time  is  the  intensity  of  light, 

(72)        •r-= 


Hence,  the  radiation  pressure  on  a  normal  surface  is  equal 
to  the  intensity  of  light  divided  by  the  velocity  of  the  prop- 
agation. 

If  the  light  falls  upon  a  perfectly  reflecting  surface  the  value 
of  the  radiation  pressure  and  the  intensity  will  be  double  the 
amount  of  that  which  it  exerts  when  falling  upon  a  perfectly 
absorbing  surface. 

Nichols  and  Hull  find  that  the  pressure  of  the  radiation  of 
sunlight  is  equal  to 


p  =  y.o        . 

cm.2       cm.3     cm.  sec.2 

The  following  formulas  define  many  of  the  relations  that 


THERMODYNAMIC   PROCESSES   IN   SOLAR  ATMOSPHERE          33 

exist  between  the  radiation  terms  and  the  thermodynamic  terms 
in  gaseous  atmospheres.     (Heaviside  and  others.) 


The  Mean  Flux  of  Energy 

K 

The  volume  density  of  radiation  =  w  =  J-f-H  =  —  ~  £02  + 

O  7T 

—  .     In  a  perfectly  reflecting  enclosure  the  total  energy  re- 

STT 

mains  a  constant.  Plane  waves  would  run  back  and  forth  be- 
tween two  parallel  plane  walls  without  losing  the  form  of  perfect 
periodic  waves,  H  =  HQ  cos  [«$  —  #),  but  if  the  walls  are  ir- 
regular in  form,  the  train  of  waves  takes  on  irregularities  which 
become  very  complex.  There  is  an  average  irregularity  which 
has  a  regular  effect,  and  this  is  constant  as  long  as  the  total  inner 
energy  remains  constant.  Insert  a  plane  area  into  the  enclosure 
at  any  place.  If  the  energy  u  moves  in  one  way  with  the  velocity 
c  the  flux  would  be  W  =  u  c;  but  this  would  produce  an  accu- 
mulation, so  that  an  equal  amount  f  c  u  moves  in  each  direction 
through  the  area  A  . 

Since  the  rays  in  the  enclosure  take  all  directions  of  6  to  the 
normal,  it  follows  that  the  integral  of  u  cos  0  over  a  hemisphere 
gives  the  flux  W. 


(73)    TT  K  =    j      j  -  sin  ed  e  d*  =  i  c.u 

4  TT  =  the  area  of  a  'sphere  of  unit  radius,  and  sin  0  d  0  dtp 

is  the  area  of  an  element  of  the  surface. 
The  specific  intensity  of  the  flux  over  the  unit  area  in  the 
unit  time  =  \  u  X  c. 


The  energy  per  unit  volume  u  —  —  —  or  -  =  —  —  or  - 

4;r         4;r         8?r         8?r 

if  H  .  E  are  average  values  and  HQ  E0  maximum  values  in  the 
plane  wave  front.  The  energy  that  crosses  the  unit  surface 
perpendicular  to  the  line  of  propagation  is  equal  to  the  amount 
contained  in  a  volume  having  the  unit  area  for  base  and  the 
distance  c  t  for  height. 


34  A   TREATISE   ON  THE   SUN'S   RADIATION 

The  Mean  Pressure  of  Radiation 

The  electric  and  magnetic  stresses  unite  to  form  a  pressure 
along  the  ray  which  is  equal  to  the  total  inner  energy  u,  with 
no  pressures  perpendicular  to  the  line  of  propagation,  that  is,  in 
the  plane  of  E  H.  The  ray  in  the  enclosure  takes  on  all  direc- 
tions, and  the  radiant  pressure  becomes  like  a  hydrostatic  pres- 
sure. If  the  energy  in  the  ray  u  goes  in  one  direction  through 
the  area  A  the  pressure  is  p  —  u;  if  it  makes  the  angle  0  with  the 
normal  u  becomes  u  cos  0,  and  p  becomes  u  cos2  0  on  A .  For  the 
average  pressure  in  all  directions,  we  have,  over  the  whole  sphere, 


Emission  and  Temperature,  Stefan's  Law 

When  the  boundary  of  the  enclosure  is  not  perfectly  reflecting 
(black),  but  partly  absorbing  (gray),  the  temperature  and  the 
second  law  of  thermodynamics  enter  into  consideration. 

Take  a  piston  with  one  head  A  fixed  and  another  B  movable. 

1.  Put  B  upon  A  so  that  the  space  is  nil  and  A  is  maintained 
steadily  at  the  temperature  T  in  a  cyclic  process. 

2.  Move  B  to  the  distance  h,  and  we  have,  with  u  p, 
(Wi  —  WQ)  =  p  h  =  the  work  of  expansion. 

(Ui  —  Z70)  =  u  h  =  the  change  in  the  inner  energy. 

(Q1   -  Qo)   =  (Wi  -  TFo)  +  (Ui  -  tfo)  =  (p  +  u)  h  =  the 

/I  \        4 

heat  lost  by  A  to  the  enclosure  =  {-^  u  +  u)  =  —  u  h. 

^  o  '          o 

3.  Fix  the  piston  B  and  let  the  cylinder  cool  down  to  0°. 
All  the  inner  energy  (Ui  —  U0)  =  u  h  goes  out  through  A. 

4.  Push  B  back  to  A  and  then  raise  A  to  the  temperature 
T.     Integrate  by  the  second  law  of  thermodynamics, 

d  Q        rTldQ    j~  Substitute  Q  =  4 «, 

T'  =  JQ     TdT'-' 


THERMODYNAMIC  PROCESSES  IN  SOLAR  ATMOSPHERE         35 

u  _      CT   1  d  u  Differentiate  and  multiply 

--    ~  J  Q 


1  ^Jf      JL  JL  d  u      j£  u_        4u      d  u        du  _      dT 
fdT~''~3TdT~~3T2'       ~T~~Tf'       ~u~         ~f~' 

(75)  log  u  =  4  log  r  +  log  a,      u  =  aT*.       p  =  -^  a  T*. 


Summary  of  the  Formulas  of  Radiation 

The  following  summary  of  the  formulas  and  the  dimensions 
of  the  terms  in  radiation  will  be  found  convenient  for  general 
reference. 

Planck's  Radiation  Terms 
(76)     Spatial  Density.     u  =  —  K  =  —  K  =  aT4  =  —  *  T*  = 

00  0 

~     _        -    _     .     M 
6  -p  = 


(77)     Auxiliary.  K  =  K  +  Ki  =  2  K  for  polarization 

along  two  axes. 


cK 


c     j  c  d\ 

since  v  =  —  .  a  ^  =  --  . 
X'  X2 


(78)     Specific  Intensity.  K  =  2K  =  ^-u  =  -^-aT4  =  —  T4 


j^p  =  ~f  =  ^rsT  =  LT-  I  ^ 

47T  lO7TZ  lOTT 

11  4 

(79)     Pressure  of  Radiation.  p  =  —u  =  —aT*  =  —<rT*  = 

o  o  o  C 

5T  PVu  PU        47T_         7       f  M 


4    w         4  4  />        16T  , 

(80)     Entropy.  5  =  — —  =  — ar3  =  -y:  =  ^-pj  K 


•T'    LL 


36  A   TREATISE   ON   THE   SUN'S   RADIATION 

(81)     Entropy  Radiation.    L  =  -j-  a  T*  =  — -  s  =  K/T  = 


47T  167T 

_£_  JL 

47r  m   T'     LPdeg 


_c_^_u_    r    M    -i 

'    LPdegJ* 


(82)     Radiation  Energy.      7  =  -p  «  =  -    -  K  =  -    -K  = 

o  o  C  o  C 

c_F     r  M 
2' 


(83)     Mechanical  Force.    F  =  Fe  +  Fm  =    -         /. 


(84)     Radiation  Intensity.     EA  =  —  -  =  ---  —  uv  =  ^r-r  —  7, 

A  O  7T  A  6Zi  IT"    A 


(85)    Monochromatic  Spectrum  lines.    «„  =  -p  «»  f  »*  V  = 

>^  <f  (if  V))  ,  Wien's  Displacement  Law. 

(c2  .xS  — 

-jp-J  .     Intensity.   K,  = 

*  P  (I)  .   .If.  7 
."    ^2v  J  •'        *- 


(87)     Specific  Intensity.    Kv  =  ~~  =  ^~  uv  = 


\  2 


O  7T 
1 


(88)     Spatial  Density.    uv  =  —  K,  =  —  Kv  =  r~ 7V  = 

C  C  4  7T 


he 


THERMODYNAMIC  PROCESSES  IN   SOLAR  ATMOSPHERE         37 

(89)     Radiation  Energy.     /„  =  — —  Kv  =  -     -  Kv  =  -  -  uv  = 

o  C  6  C  6 


(90)     Radiation  Intensity.     Ex  =  -^-»  K  =  -~  =  -  —  — 

A   A  A  O  7T  A 


3       J?  _   £^ 

x   o      •*  V     -        »    c 


-1 


(91)     Unpolarized  Spatial  Density.      —  E\  =   —  Kv  =  • — ^" 

C  A  A 


Planck  bases  all  of  these  formulas  upon  strictly  adiabatic 
conditions,  assuming  a  cylinder  with  perfectly  reflecting  sides 
or  impermeable  walls.  Hence  they  do  not  apply  to  free  at- 
mospheres without  modifications. 

Tables  of  Radiation  Constituents  in  the  Volume 

The  following  tables  give  the  values  of  the  terms  that  are 
concerned  with  radiation  in  the  different  volumes  of  the  sun's 
and  the  earth's  atmospheres,  taking  them  at  rest.  When  multi- 
plied by  the  velocity  of  light  they  become  radiation  fluxes. 
They  are  valuable  in  tracing  the  intrinsic  changes  from  level 
to  level  within  the  atmospheres,  due  to  the  many  physical 
processes  at  work.  The  electromagnetic  plane  waves  trans- 
port the  energy  that  is  finally  able  to  escape  from  the  sun,  after 
the  original  radiation  at  the  levels  ZR  has  been  depleted  by  an 
equivalent  of  about  1.87  gr.  cal./cm.2  min.  in  the  higher  strata 
of  the  solar  envelope. 


38 


A   TREATISE   ON   THE   SUN  S   RADIATION 


c-  <- 


fc     S 
0 


ee 


3    Ji 

O         3 


s 


-T  1 


ft 


£ 


>    H    in 


c 


§   1 


THERMODYNAMIC   PROCESSES   IN   SOLAR  ATMOSPHERE          30 


1,       >       >        ^        ^      v      V"V__l!_>:        ** 


s 

CJ 

\ 

A 

! 

S1 

i 

tD   bC 
0)    u 
*O    Q^ 

bi   ^      ^ 

<u       &          5                b* 

^     B?       ii            fc 

u 

d      ff  2? 

O                        o               ^ 

t 

] 

i 

s 

1    -   1 

1         „         g               j 

1 

i-            2     S 

\ 

|, 

EC 

en    S 

n! 

i' 

"0  § 

CJ     I-N 

SP«.          N.   *        bi) 

«    .    8   .    8  d     "> 

U    v-             u              C 
en    b/j          bfi           O 

w. 

u 

flj                   1-t         1-1 

•  -rt"  '8    s 

s  s  s  ^     3 

U     U        (O         ^T1              *T 

c 

:>           C 

5           C 

5           C 

5            C 

5          0          0       0          C 

D            C 

D       6       b       b 

C 

<u 

o; 

J 

| 

r- 

1      1 

.  J22 

o           \S 
Oj            0 

tn           i> 

"gl 

V 

1 

g                               »—  > 

. 

' 

i 

„  § 

1 

0)    5t 

1         II         S           ^ 

•         M 

#  s      g  ,  -S     . 

1 

^^                 (1) 

',          x     J 

¥ 

&u 

*;•  # 

£  $ 

w      b^ 

"V$    8!$  Hi     ri 

^M 

g£     S       O        i—  1            i—  l 

'2 

* 

&£ 

^ 

S^^ 

^15       13        S     ** 

§^ 

"§' 


I 

j 

^  \ 

CO  1  C^         - 

^      s 

II       » 

j^ 

fX^  | 

t* 

•       tn           g 

S«4                       ^ 

i 
ft 

CO  1 

1 

b«       C           £3          »|j 

•8    3      §      •- 

^           3            N 

ol             CT           rr 

^  1 

&     fc 

R 

C 

0 

K 


w  p<j 


8      J 


S, 
w 
W 
S 


3 


^          a> 


§     1 


O  T-l  TH 


40 


A    TREATISE    ON   THE    SUN  S    RADIATION 


;- 

*V           >-           V            " 

-     " 

1* 

Ti 

bfl 

bfl 

cu 

£? 

0) 

!        _r 

1 

s 

c/5 

O 

'"1 

u  s?  "s  &?  a  S*  s    i 
^s  fc  §  «  8  *  u 

bb 

cu 

bfl 

o 

0)         ||               II               II 

cu 

1 

U 

"O 

-o 

bfl 

CJ                     CJ                     O                     (J       >-< 

8"  . 

35    sj 

y 

d^  i6§6d 

6 

. 

C 

5           O           O           O           C 

5           C 

> 

% 

o 

CU 

0)                                                   CJ 

bfl 
cu 

H 

1 

-o                              3 

0 

•9 

§ 

CO 

j 

*4U  •*]*  4 

J2 

M 

* 

_    1 

^  A  "S  J,  "S  ^   8      ' 

3 

o 

8 

<j 

S 

^     II         II 

sP 

1 

£ 

I 

lililill 

"O 
en    O 

H 

s      s      s      s 

g^ 

S 

C/3 

a 

1 

bfl 
cu 
"O 

bfl 
0) 

-o 

u 

3 

g 

^^     £H^     ^^     ^^ 

^5 

k 

s 

| 

"S1                                   S 

,           ' 

i 

0 

N 

<o 

1 

"*                Tj< 

CO  00 

CO 

'  f 

i                                       «                   ||                J 

I 

f                               ^       ^          ^                  „ 

»           CJ 

.5 

•  < 

\                                   1           Jbl             H« 

3 

j              T                                 CJ                O 

0 

1 

.1 

< 

^               ^         J* 

$              <$              V         M              >• 

ll 

C 

.2 

0 

I    *    1   fe1   i 

.2 

;    I 

\ 
( 

1 

^ 

:         >>        a      fc  c    G 

•                -in                  rj              C      O         CU 

3         2         B    .  o  TS  • 

^      S      -5    .a  .2  .a 

i  3  i  in 

>        p4      ^        en 

black  radi 
Mechanical 

1 
1 

I 


M 


bfl 


S 


vo|, 

II 

b 

j2 
_c 

c  s 


=  1 

«  ^ 

o  ^ 

u 


C  CJ 

I       g 


w 


THERMODYNAMIC   PROCESSES   IN   SOLAR  ATMOSPHERE 

TABLE  4 

TABLE  OF  ASTRONOMICAL  CONSTANTS 
Meter-Kilogram-Second  (M.  K.  S.)  System 


41 


Number 

Logarithm 

r 

=  Mean  radius  of  the  earth 
(Bessel's  spheroid)  

6370191  m. 

6  8041525 

R 

R/r 
D 

=  Mean  radius  of  the  sun, 
(Auwer'sdiam.31'59.26") 
=  Ratio  of  the  radii  R/r.  .  . 
=  Distance  of  the  sun  to 
the  earth              

694800800  m. 
109.071 

149  340  870  000  m 

8.8418603 
2.0377078 

11  1741786 

RID 

=  Ratio  R/D  

0  0046525 

7.6676817  —  10 

(R/DY 

/A2 
=  Ratio  I  —  1   . 

0  000021645 

5.3353634  —  10 

P 

\D) 

=  Parallax  of  the  sun,  p  = 
tan  (r/D)  . 

8"  80756 

5  6299739  —  10 

P 

=  Parallax     of     the     sun 
(Newcomb) 

8  7965 

S/Si 

=  Ratio    of    the    surfaces, 
S/Si  =  (109.  071)2  

11896  4 

4  0754156 

V/Yi 

=  Ratio   of    the    volumes, 
V/Vi  =  (109.  071)3  

1297548 

6  1131234 

Pa 
Ps 

m 

=  Average  density  of  the 
earth  (Harkness)  
=  Average  density  of  the 
M  fry 

SUn  =  m(R)Pa'-' 

=  Mass  of  the  earth  in  kilo- 
grams              . 

5.576kil./m.3 
1.  43287  kil./m.3 
6.0377Xl024kil. 

0.7463230 
0.1562056 

24.780872 

M 

=  Mass  of  the  sun  in  kilo- 
grams 

2.0132X1030  kil. 

30.303878 

M/m 
M/m 
G 
Go 

7 
k 

=  Ratio,  mass  of  the  sun 
to  the  earth,  M/m  
=  Ratio,  mass  of  the  sun 
to  the  earth  (Newcomb)  . 
=  Acceleration  per  second 
at  surface  of  the  sun  
=  Acceleration  per  second 
at  surface  of  the  earth.  .  .  . 
=  Ratio    of    the   accelera- 

M  r2 
tions,  G/go  =  —  —  
m  K 

The  gravitation  constant, 
k  =  g0  rz/m  

333431 
333432 
274.843  m./sec. 
9.8060  m./sec. 

28.028 
6.5906X10~U 

5.523006 
5.523008 
2.4390844 
0.9914920 

1.4475904 
9.818925-20 

V 

=  Velocity  of  the  earth  in 
its  orbit 

29806  6  m./sec. 

4.474313 

18.  52  12  miles/sec. 

1.267670 

42 


A   TREATISE   ON   THE   SUN  S   RADIATION 


TABLE  4^-Continued 

TABLE  OF  ASTRONOMICAL  CONSTANTS 

Meter-Kilogram-Second  (M.  K.  S.)  System 


Number 

Logarithm 

/ 

=  Acceleration  at  the  dis- 

tance of  the  earth  =  -—  . 

0.0059491  m./sec. 

7.774446-10 

Check,/  =  -(^Y  go. 
m   \D  J 

7.774446-10 

S 

=  Rate  at  which  the  earth 

falls  toward  the  sun  =  %f 

0.0029746  m./sec. 

7.473416-10 

1" 

(second  of  arc)  = 

radius  of  sun  in  kilometers 

radius  of  sun  in  seconds  of  arc 

724.030  kil. 

2  .  8597565 

The  astronomical  constants  given  in  Table  4  will  be  found 
useful  in  reference  to  the  general  relations  between  the  earth 
and  the  sun,  and  these  data  will  be  employed  in  this  Treatise. 


CHAPTER  II 
Computation  of  the  Thermodynamic  Terms 

General  Remarks 

HAVING  obtained  the  data  for  one  point  of  thermal  equi- 
librium in  the  gas  under  consideration,  it  is  necessary  to  proceed 
by  the  method  of  trials  from  level  to  level,  both  above  and 
below  that  point.  In  order  to  accomplish  this  purpose,  the 
formulas  (l)  to  (5)  are  worked  through  in  the  non-adiabatic 
form;  then  the  terms  in  the  gravitation  formula  are  computed: 

(43)   G.  (z.  -  Z0)  =  -  Z=Z  -  (Cpa  -  CpJ  (T.  -  T,)_+  A  G\. 

p"> 

fe  -  *)• 

Take  a  series  of  values  of  T,  as  7\,  T2,  T3,  .  .  .  Tw,  and 
compute  the  residuals  A  Go  (zi  —  z0) .  There  is  one  definite  value 
of  T  which  will  make  the  residual  a  minimum.  As  Tn  approaches 
this  value  the  residuals  diminish;  as  Tn  passes  the  minimum 
residua]  these  increase.  After  some  practise  with  the  curve  of 
the  residuals  the  value  of  T  may  be  readily  found  approximately 
for  each  new  level,  and  as  the  curves  for  P,  p,  R,  T,  develop  it 
is  easy  to  project  T  quite  accurately.  The  labor  of  working 
through  the  first  four  elements,  hydrogen  (diatomic),  calcium, 
cadmium,  mercury,  was  very  great,  but  the  curves  soon  re- 
vealed the  law  of  the  temperature  distribution  with  the  height 
above  the  photosphere,  so  that  the  other  elements,  hydrogen 
(monatomic),  helium,  carbon,  zinc,  were  computed  with  very 
little  need  of  extensive  trials.  The  number  of  trial  computations 
was  from  six  to  ten  for  hydrogen  (diatomic),  but  it  diminished 
to  less  than  two  in  the  last  elements  that  were  tried.  It  is  now 
easy  to  assign  the  temperatures,  pressures,  densities,  and  thermal 
efficiencies  for  any  gas  quite  closely  without  the  use  of  the 
complete  set  of  formulas.  An  examination  of  the  curves  on 

43 


44 


A   TREATISE   ON   THE   SUN  S   RADIATION 


COMPUTATION  OF  THE  THERMODYNAMIC  TERMS 


46 


A   TREATISE    ON   THE    SUN'S    RADIATION 


COMPUTATION  OF  THE  THERMODYNAMIC  TERMS 


47 


§ 
a 


48 


A   TREATISE   ON  THE   SUN'S  RADIATION 


COMPUTATION   OF   THE   THERMODYNAMIC    TERMS 


49 


50 


A   TREATISE   ON   THE   SUN  S   RADIATION 


COMPUTATION  OF  THE  THERMODYNAMIC  TERMS 


51 


52 


A  TREATISE   ON  THE   SUNJS  RADIATION 


Figs.  2-9  shows  that  they  are  all  constructed  on  the  same  model, 
while  Figs.  10-13  indicate  that  they  form  families  of  curves 
which  are  interrelated  by  hyperbolic  conditions  with  the 
parameter  m. 

The  constitution  of  the  atmospheres  of  the  several  gases  on 
Z 


25000 


20000 


15000 


10000 


5000 


-5000 


-10000 


He 


Top  of  the  nnercorom 


4n 


Top  of  the 


Jhromosphe 


2000 


4000 


6000 


8000 


10000 


13000 


FIG.  10.     T.    Temperature   in   Absolute    Centigrade    Degrees. 
Z  =  height  in  kilometers  from  the  photosphere. 

the  sun  is  exactly  the  same  in  principle  as  was  found  in  the 
atmosphere  of  the  earth.  This  is  seen  by  comparing  the  curves 
of  Fig.  1,  which  summarizes  the  terrestrial  data  for  the  balloon 
ascension,  Uccle,  September  13,  1911.  Compare  Bigelow's 
Treatise,  Tables  96,  97;  In  the  case  of  the  earth's  atmosphere, 
the  lower  adiabatic  section  is  not  very  clearly  developed  at  the 
latitude  of  Belgium,  because  the  temperature  gradients  in  the 


COMPUTATION  OF  THE  THERMODYNAMIC  TERMS 


53 


lower  levels  do  not  agree  with  the  adiabatic  rate.  This  would, 
however,  be  found  generally  for  the  balloon  ascensions  made  in 
the  Tropics.  The  isothermal  layers,  12000  to  37000  meters,  and 
the  non-adiabatic  layers,  38000  to  90000  meters,  are  clearly 
determined.  The  peculiar  secondary  curvature  in  the  gas 


20,000   H 


15,000 


10,000 


5000 


-5000 


-10,000 


0  20  40  60  80  100  120 

FIG.  11.     A.  Pressure  in  Terrestrial  Atmosphe 


efficiency  R,  which  occurs  in  the  levels  where  the  temperature 
gradient  is  at  its  greatest  value,  is  found  in  all  the  atmospheres. 
In  the  case  of  the  sun  there  were  from  50  to  70  points  deter- 
mined on  each  of  the  curves  of  P,  p,  R,  T,  from  deep  within  the 
adiabatic  region  to  the  vanishing  plane  of  the  gas.  An  inspection 
of  Figs.  2-9  shows  clearly  that  the  gases  Hi,  H2,  He,  C,  Ca,  Zn,  Cd, 


54 


A  TREATISE   ON  THE   SUN'S   RADIATION 


Eg  differ  from  one  another  chiefly  in  the  value  of  the  vertical 
scale,  or  depth,  within  which  the  gas  develops  its  distribution. 
The  original  computations  in  logarithms  check  very  closely 
throughout;  these  are  reserved  for  another  publication.  In 
order  to  make  clear  the  series  of  fundamental  laws  of  distribution, 


15,000 


10,000 


5000 


-5000 


-10,000 


0.100 


0.500 


o.uoo 


0.200       0.300       0.400 

FIG.  12.     p.  Density. 

kil°grams  ;    £(M.K.S.) 
meter2          L3 

the  values  of  T,  P,  A,  p,  R  have  been  interpolated,  where  neces- 
sary, so  that  the  values  are  here  given  on  the  same  selected 
levels  for  each  of  the  gases.  As  all  of  the  terms  will  be  studied 
in  detail,  it  is  not  now  necessary  to  explain  further  the  Figs.  2-9. 
It  will  be  convenient  to  give  at  this  point  Table  5  of  the  atomic 
and  molecular  weights  of  all  the  chemical  elements,  together 
with  the  approximate  order  of  brightness  of  occurrence  of  those 
identified  upon  the  sun  generally,  but  in  spots  and  faculae 
especially. 


COMPUTATION   OF    THE    THERMODYNAMIC    TERMS 


55 


a  =  the  atomic  weight  referred  to  hydrogen,  H  =  1.00, 

(0  =  15.88). 

m  =  the  corresponding  molecular  weight  for  some  ele- 
ments. 
ai  =  the  atomic  weight  referred  to  H  =  1.008  (O  =  16.00). 


Z 

20000 


19000 


He 
10000 


5000 
C 


c 

Zn 

CdQ 

Hg 

He 


-5000 


25000 


\ 


He 


Photosphere 


50000  75000  100000 

FIG.  13.     R.  Gas-Coefficient. 
velocity2 


136000 


150000 


degree   '    jPdeg. 


(M.  K.  S.) 


m-i  =  the  corresponding  molecular  weight. 
O  B  =  the  order  of  brightness  in  the  solar  spectrum  (l  to  36). 
O  L  =  the  order  of  the  number  of  lines  in  the  solar  spec- 
trum (1  to  36). 
O  S  =  the  number  of  lines  of  the  element  in  sun-spots. 


56  A   TREATISE   ON  THE   SUN7S   RADIATION 

TABLE  5 

THE  ATOMIC  AND  MOLECULAR  WEIGHTS 
The  Occurrence  of  the  Chemical  Elements  on  the  Sun 


Element 

5 

H  =  1.00 

m 

« 
H=  1.008 

mi 

OB 
Order 

OL 

Order 

OS 
No. 

OF 
No. 

Aluminum.  .  . 
Antimony  .  .  . 
Argon  .  . 

Al 
Sb 
A 
As 
Ba 
Bi 
B 
Br 
Cd 
Cs 
Ca 
C 
Ce 
Cl 
Cr 
Co 
Cb 
Cu 
Dy 
Er 
Eu 
F 
Gd 
Ga 
Ge 
Gl 
Au 
He 
Ho 
H 
In 
I 
Ir 
Fe 
Kr 
La 
Pb 
Li 
Lu 
Mg 
Mn 
Hg 
Mo 
Nd 

26.88 
119.24 
39.56 
74.37 
136.28 
206.35 
10.91 
79.29 
111.51 
131.57 
39.75 
11.91 
139.14 
35.18 
51.59 
58.50 
92.76 
63.07 
161.21 
166.37 
150.79 
18.85 
156.05 
69.35 
71.92 
9.03 
195.64 
3.97 
162.20 
1.000 
113.89 
125.91 
191.57 
55.40 
82.26 
137.90 
205.56 
6.88 
173.61 
24.13 
54.49 
198.47 
95.24 
143.16 

27.10 
120.20 
39.88 
74.96 
137.37 
208.00 
11.00 
79.92 
112.40 
132.81 
40.07 
12.00 
140.25 
35.46 
52.00 
58.97 
93.50 
63.57 
162.50 
167.70 
152.00 
19.00 
157.30 
69.90 
72.50 
9.10 
197.20 
4.00 
163.50 
1.008 
114.80 
126.92 
193.10 
55.84 
82.92 
139.00 
207.20 
6.94 
175.00 
24.32 
54.93* 
200.06 
96.00 
144.30 

•• 

39.56 

297.48 

39.88 
299.84 

9 

25 

Arsenic 

Barium  

15 
D 

24 
D 

2 

Bismuth 

Boron  .  . 

Bromine  

158.58 
111.51 
131.57 
39.75 

159.84 
112.40 
'132.81 
40.07 

27 
1 

16 

28 

11 
7 
22 
25 

26 

11 
7 
10 

5 
6 
16 
30 

Cadmium..  .  . 
Caesium  
Calcium 

60 

1 
2 

386 
118 

33 
105 

47 

*78 
66 

Carbon  

Chlorine 

70.36 

70.92 

Chromium.  .  . 
Cobalt  
Columbium.  . 
Copper  
Dysprosium.  . 
Erbium  

35 

28 

.  .  . 

•  •  • 

Europium  .  .  . 
Fluorine  

38.00 

37.70 

Gadolinium.. 
Gallium 



Germanium.  . 
Glucinum.  .  .  . 
Gold 

30 
29 

33 
32 

4 

2 

Helium  .  . 

3.97 

4.00 

•• 

3 

Holmium.  .  .  . 
Hydrogen.  .  .  . 
Indium. 

2.000 

2.016 
253.84 

3 
D 

D 
2 

21 

34 

22 
D 

D 
1 

14 
35 

Iodine  .  .  . 

251.82 

1108 

327 

24 

Iridium  

Iron  

Krypton  
Lanthanum  .  . 
Lead  
Lithium  
Lutecium.  .  .  . 
Magnesium.  . 
Manganese.  . 
Mercury  
Molybdenum 
Neodymium  . 

82.26 

82.92 

6.88 

6.94 



6 
13 
D 
20 
24 

19 
4 
D 
17 
12 

8 
167 

3 

5 
21 

'42 

198.47 

200.06 



COMPUTATION  OF  THE  THERMODYNAMIC  TERMS 

TABLE  5  (Continued} 

THE  ATOMIC  AND  MOLECULAR  WEIGHTS 

The  Occurrence  of  the  Chemical  Elements  on  the  Sun 


57 


Element 

5 

a 
H  =  1.00 

m 

ai 
H  -  1.008 

mi 

OB 
Order 

OL 
Order 

05 
No. 

OF 
No. 

Neon      .  . 

Ne 
Ni 
Nb 
M 

N 
Os 
0 
Pd 
P 
Pt 
K 

Pr 
Ra 
Rh 
Rb 
Ru 
Sa 
Sc 
Se 
Si 
Ag 
Na 
Sr 
S 
Ta 
Te 
Tb 
Tl 
Th 
Tm 
Su 
Ti 
W 
U 
V 
Xe 
Yb 
Yt 
Zn 
Zr 

19.86 
58.21 
92.76 
220.63 

13.90 

189.39 
15.88 
105.85 
30.79 
193.65 
38.79 

139.78 
224.21 
102.08 
84.77 
100.89 
149.21 
43.75 
78.57 
28.08 
107.02 
22.82 
86.93 
31.81 
180.06 
126.49 
157.94 
202.38 
230.56 
167.16 
117.76 
47.72 
183.00 
236.31 
50.59 
129.71 
172.12 
88.00 
64.85 
89.88 

19.86 

20.02 
58.68 
93.50 
220.40 

14.01 
190.90 
16.00 
106.70 
31.04 
195.20 
39.10 

140.90 
226.00 
102.90 
85.45 
101.70 
150.40 
44.10 
79.20 
28.30 
107.88 
23.00 
87.63 
32.06 
181.50 
127.50 
159.20 
204.00 
232.40 
168.50 
118.70 
48.10 
184.00 
238.20 
51.00 
130.20 
173.50 
88.70 
65.37 
90.60 

20.02 

5 

2 

251 

71 

Nickel  
Niobium  .... 
Niton(radium 
emanation) 
Nitrogen  .... 
Osmium.  .  .  . 

27.80 
31.76 

28.02 
32.00 

D 
D 
23 

D 

36 

D 
D 
18 

D 
36 

.  .  . 

.  .  . 

Oxygen  
Palladium  .  .  . 
Phosphorus.  . 
Platinum.  .  .  . 
Potassium.  .  . 
Praseodym- 
ium. 

123.16 

124.16 
39.  io 

... 

... 

38.79 

Radium  
Rhodium.  .  .  . 
Rubidium.  .  . 
Ruthenium  .  . 
Samarium  .  .  . 
Scandium.  .  .  . 
Selenium  .... 
Silicon  



31 
D 

27 
D 

84.77 

85.45 

17 

13 

45 

34 
4 

6 
2 

157.14 

158.40 

Silver     .... 

32 
4 
12 

D. 

31 
20 
23 

D 

8 
2 

Sodium  
Strontium  .  .  . 
Sulfur  

22.82 

23.00 
64.12 

63.62 

Tantalum.  .  .  . 
Tellurium.  .  . 
Terbium  
Thallium  
Thorium..  .  . 

252.98 

255.00 



D 
D 

D 
D 

... 

Thulium  
Tin.      . 

33 
10 
D 
D 
14 

34 
3 
D 
D 

8 

Titanium.  .  .  . 
Tungsten..  .  . 
Uranium  .... 
Vanadium.  .  . 
Xenon  
Ytterbium.  .  . 
Yttrium  
Zinc  
Zirconium  .  .  . 

432 

131 

176 

42 

130.20 

129.71 

64*85 

18 
26 
19 

15 
29 
9 

3 
3 

7 

19 
2 

65.37 

58 

0  F  =  the  number  of  lines  of  the  element  in  the  flash 

spectrum. 

D  =  doubtful  identification  in  the  solar  spectrum. 
The  order  of  brightness  is  taken  from  Abbot's  Sun,  page  206. 
The  number  in  sun-spots,  Astro  physical  Journal,  June,  1913, 
St.  John. 

The  number  in  the  flash  spectrum,  Astro  physical  Journal. 
March,  1915,  Mitchell. 

1  am  indebted  to  Professor  F.  W.  Clarke  for  the  revised 
values  of  the  atomic  weights,  which  are  those  of  the  Inter- 
national Committee  for  1916. 

•*: 

The  Distribution  of  the  Temperatures 

The  computations  were  conducted  in  height  by  taking  as 
0,  or  the  reference  plane,  the  point  of  equilibrium  that  was 
derived  from  the  terrestrial  data  through  the  factor  p,  but  as 
this  point  has  no  special  significance  in  solar  physics  it  has  been 
found  convenient  to  define  two  other  planes  of  reference,  (1)  the 
surface  of  the  photosphere,  and  (2)  the  bottom  of  the  isothermal 
layer.  By  means  of  the  spectroheliographic  observations  made 
at  Mt.  Wilson,  and  by  other  spectroscopic  determinations,  the 
pressure  on  the  photosphere  lies  between  5.80  and  6.00  standard 
terrestrial  atmospheres,  and  this  has  been  accepted  as  the  defini- 
tion of  the  level  of  the  photosphere.  Compare  Fig.  11,  where 
all  the  pressure  curves  pass  through  this  point,  no  matter  to 
what  heights  and  depths  the  several  pressures  extend  in  their 
distribution.  This  is  confirmed  as  correct  by  the  general  fact 
that  the  diatomic  hydrogen  vanishes  at  25000  kilometers  above 
the  photosphere,  at  the  top  of  the  inner  corona  as  observed  in 
eclipses,  that  the  top  of  the  chromosphere,  5000  kilometers,  is 
at  the  level  where  the  hydrogen  pressure  and  density  become 
very  small,  as  in  Figs.  11,  12.  The  light  gases  extend  beyond 
the  level  of  the  reversing  layer  which  is  at  400-500  kilometers, 
but  the  heavy  gases  do  not  reach  it  normally,  only  in  great 
disturbances,  and  so  do  not  generally  appear  in  the  flash  spec- 
trum of  the  Fraunhofer  lines.  The  vanishing  monatomic  molec- 


COMPUTATION  OF  THE  THERMODYNAMIC  TERMS      59 

ular  weights  range  for  the  flash  spectrum  from  60  to  80,  so 
that  helium  (4),  carbon  (as  12),  calcium  (40),  zinc  (65),  appear 
readily,  while  cadmium  (112),  mercury  (198),  are  seen  with 
difficulty.  Thus,  the  flash  spectrum  by  Adams*  gives  some- 
what the  following  distribution  of  lines  by  numbers  as  counted: 
Carbon  (12),  105;  calcium  (40),  33;  cerium  (139),  47;  cobalt 
(59),  66;  chromium  (52),  78;  iron  (55),  327;  glucinum  (9),  2; 
helium  (4),  3;  lanthanum  (138),  24;  magnesium  (24),  5;  man- 
ganese (54),  21;  sodium  (23),  6;  neodymium  (143),  42;  nickel 
(58),  71;  scandium  (44),  34;  silicon  (28),  4;  strontium  (87),  2; 
titanium  (48),  131;  vanadium  (51),  42;  yttrium  (89),  19;  zinc 
(65),  2;  as  contained  in  the  tables.  The  exceptions  are  cerium 
(139),  lanthanum  (138). 

Table  6  contains  the  temperatures,  and  it  shows  clearly 
the  distribution  of  the  isothermal  temperature,  deep  for  hydrogen 
and  shallow  for  mercury,  with  non-adiabatic  layers  above  and 
adiabatic  layers  below  it.  It  is  seen,  from  Fig.  11,  that  the 
pressures  have  one  point  in  common  in  passing  the  plane  of  the 
photosphere  at  6  atmospheres,  and  from  Fig.  10,  that  the  tem- 
peratures have  one  nearly  vertical  line  in  passing  the  photosphere, 
namely,  the  isothermal  layer,  having  a  temperature  for  each  gas 
of  about  7650°  at  the  bottom,  and  of  about  7710°  at  the  top 
of  this  characteristic  layer.  The  means  refer  to  the  tempera- 
tures in  the  isothermal  region  to  the  left  of  the  broken  line  of 
separation.  The  non-adiabatic  region  is  above  this  line  and  the 
adiabatic  region  is  below  it.  The  mean  value  of  the  temperature  of 
the  isothermal  layer  is  about  7687°  absolute  Centigrade.  Compare 
the  Table  6  and  Figs.  2  to  9.  These  figures  were  constructed 
on  different  convenient  vertical  scales  of  heights  for  ordinates, 
and  with  various  abscissas  for  P,  p,  R,  T,  respectively,  in  order 
to  show  the  interrelation  of  these  quantities  for  each  gas;  the 
Figs.  10  to  13  bring  together  on  the  same  vertical  scale  all  the 
elementary  gases,  for  temperature,  pressure,  density,  gas  co- 
efficient, in  order  to  determine  the  general  laws  of  their  dis- 
tribution. It  is  easily  found  that  the  equilateral  hyperbola, 
x  y  =  c,  is  fundamental  and  since  this  passes  over  into  the 

*  Astroph.  Journ,,  March,  1915. 


60 


A  TREATISE   ON  THE   SUN'S   RADIATION 


TABLE  6 
THE  TEMPERATURES  OF  THE  DIFFERENT  ELEMENTS  ON  THE  SAME  LEVELS 


z 

Kilometers 

Hi 
1.00 

H-2 

2.00 

He 
4 

12 

Co 
40 

Zu 
65 

Cd 
112 

Hg 
198 

Means 

25000.... 
20000.... 
15000... 
10000.... 
5000.... 

1000... 
900.... 
800.... 
700.... 
600.... 

500.... 
400.... 
300.... 
200.... 
100.... 

90.... 
80.... 
70.... 
60.... 
50.... 

40.... 
30.... 
20.... 
10.... 

Photos.  0.  . 

-   10.... 
-   20.... 
-   30.... 
-   40.... 

-   50... 
-   60.... 
-   70.... 
-   80.... 
-   90.... 

-  100... 
-  200.... 
300 

3320 
4300 
5370 
6600 
7550 

0 
955 
2055 
3495 
5890 

Topo 

f  the  Inn 

er  Coron 

a 

0 
740 
3780 

'6 

..  .„ 

300 
900 
1500 
2200 
2950 

3750 
4700 
5750 
6800 
7500 

7600 

Top  o 

Chromos 

phere 

7688 
7680 
7685 
7691 
7695 

7698 
7702 
7705 
7698 
7697 

7696 
7681 
7694 
7701 
7705 

7696 
7696 
7699 
7698 

7693 

7691 
7690 
7688 
7686 

7684 
7687 
7685 
7683 
7687 

7690 
7686 
7684 
7684 

7682 
7684 
7677 
7674 
7672 

7669 
7670 
7681 
7661 
7661 

7668 
7678 
7686.7 

7696 
7710 
7708 
7706 
7705 

7704 
7703 
7702 
7702 
7701 

7701 
7701 
7701 
7701 
7701 

7701 
7701 
7701 
7701 

7701 

7701 
7701 
7701 
7701 

7701 
7701 
7701 
7701 
7701 

7700 
7699 
7698 
7697 

7696 
7695 
7694 
7694 
7693 

7692 
7691 
7686 
7681 
7676 

7671 
7666 

7680 
7679 
7678 
7677 
7676 

7675 
7674 
7673 
7672 
7671 

7671 
7671 
7671 
7671 
7671 

7670 
7670 
7670 
7670 

7670 

7670 
7670 
7670 
7670 

7669 
7669 
7669 
7669 
7669 

7669 
7668 
7667 
7666 

7665 
7664 
7663 
7662 
7661 

7660 
7662 
7676 
7640 
7651 

7665 
7690 

7600 
7650 
7670 
7690 
7705 

7715 
7714 
7713 
7712 
7711 

7710 
7710 
7709 
7709 
7708 

7708 
7707 
7707 
7706 

7705 

7705 
7704 
7704 
7703 

7703 
7702 
7702 
7701 
7700 

7699 
7698 
7697 
7696 

7695 
7693 
7691 
7689 
7687 

7685 
7665  | 
7680  | 
9451 
11354 

13255 
15157 
17059 
18961 
20863 

22765 
24667 

6450 
6700 
7000 
7250 
7500 

7615 

0 
640 

1790 
3120 
4480 
5940 
7500 

7600 

Revers 
0 
1200 
2900 
5700 

6000 

ing  layer 

7715 
7710 
7705 
7704 

7703 
7702 
7701 
7700 
7699 

7698 
7698 
7697 
7696 

7695 

7694 
7693 
7692 
7691 

7691 
7690 
7689 
7688 
7687 

7686 
7685 
7680 

0 
2500 
3950 

4450 
4950 
5450 
5900 
6350 

6750 
7100 
7400 
7600 

7650 
7690 
7705 
7715 

7713 
7711 
7709 
7707 

7705 

7703 
7701 
7699 
7697 

7695 
7693 
7691 
7689 
7687 

7685 
7665 
7676 

7650 
7690 
7720 
7735 

7730 
7725 
7720 
7718 

7715 

7711 
7707 
7703 
7699 

7695 
7691 
7687 
7683 
7679 

7675 

7700 

6600 
6950 
7300 

7475 

7650 
7670 
7690 
7687 

7684 

7680 
7677 
7673 
7669 

7665 

7670 

7667 
7663 
7659 
,  7655 

7651 

7660 
7656 
7652 
7686 

7720 

7800 
8550 
9494 
10436 

11381 
20816 
30251 
39686 

49121 
58556 
67991 
77426 
86861 

96296 
190648 
285000 
379352 
473704 

568056 
662408 
756760 
851112 
945470 

1039822 
1134174 

12742 
18045 
23348 

28651 
33954 
39257 
44560 
49863 

55166 
108193 
161220 

214247 
267274 

320201 
373228 
426255 
479282 
532309 

585336 
638363 

-  400.... 

-  500.... 
-  600... 
-  700.... 
-  800... 
-  900.... 

-  1000.... 
-  2000.... 
-  3000.... 
-  4000.... 
-  5000.... 

-  6000.... 
-  7000.... 
-  8000.... 
-  9000.... 
-10000.... 

-12000.... 
-14000.... 

7675 

7670 
7665 
7658 
7651 
7646 

7640 

11974 
17680 
23387 
29094 

34801 
40508 
46215 
51922 
57629 

63336 
69043 

8651 

10553 
12455 
14357 
16259 
18161 

20063 
39083 
58103 
77123 
96143 

115163 
134183 
153203 
172223 
191243 

229283 
267323 

13797 

16881 
19965 
23049 
26133 
29217 

32301 
63140 
93979 
124818 
155657 

186496 
217335 
248174 
279013 
309852 

340691 
371530 

7661 
7656 
7651 

7690 

8000 
8500 
9187 

10561 
11934 

8476 

-47.554-686.90-1902.17-5706.5-19021.7  -30839.3  -53027.5  -94352.0 
Adiabatic  gradient  per  1000  kilometers. 


COMPUTATION  OF  THE  THERMODYNAMIC  TERMS 


61 


logarithmic  law  of  decrease  or  depletion,  this  distribution  con- 
forms to  the  natural  law  of  physical  conditions. 

The  bottom  of  the  isothermal  layer  is  nearly  the  same  as 
the  top  of  the  adiabatic  strata,  making  some  allowance  for  the 
transition  from  the  adiabatic  to  the  isothermal  conditions;  the 
top  of  the  isothermal  layer  is  nearly  coincident  with  the  bottom 
of  the  non-adiabatic  strata.  The  adiabatic  lines  of  pressure 
are  nearly  parallel  to  the  plane  of  the  photosphere,  especially  for 
the  heavy  gases;  the  non-adiabatic  lines  of  pressure  are  nearly 
vertical  to  the  plane  of  the  photosphere;  the  isothermal  layer 
of  temperature  occurs  within  the  layers  where  the  pressure  is 
changing  rapidly  from  very  large  to  very  small  gradients.  The 
density  line  for  the  light  gases  in  the  adiabatic  layers  is  nearly 
vertical,  as  hydrogen,  but  horizontal  for  the  heavy  gases;  the 
gas  coefficient  has  a  very  great  range  for  hydrogen,  but  contracts 
to  a  very  short  range  for  mercury.  It  follows  that  the  solar 
atmospheres  constitute  a  very  perfect  thermodynamic  engine 
in  the  adiabatic  and  isothermal  layers,  but  one  of  less  efficiency 
in  the  non-adiabatic  region,  which  can  be  reduced  to  correspond- 
ing secondary  adiabatic  and  isothermal  components. 

The  relations  between  the  height  z,  the  atomic  weight  m, 
and  the  temperature  T,  are  very  interesting  and  instructive,  as 
can  be  seen  in  the  following  tables: 

TABLE  7 
BOTTOM,  TOP,  AND  DEPTH  OF  THE  ISOTHERMAL  LAYER 


Take  m    =  the  atomic  weight,  or  molecular  weight  (monatomic). 
ZA    —  the  bottom  of  the  isothermal  layer 
zi     =  the  top  of  the  isothermal  layer 
zi  —  ZA    =  the  depth  of  the  isothermal  layer 

Element 

m 

ZA 

zi 

ZI-ZA 

Hydrogen  Hi 

1.00 
2.00 
4.00 
12.00 
40.00 
65.00 
112.00 
198.00 

-12000 
-  6000 
-  3000 
-  1000 
-     300 
-     185 
-     107 
-       60 

+3600 
+  1800 
+  900 
+  300 
+     90 
+     55 
+     32 
+     18 

15600 
7800 
3900 
1300 
390 
240 
139 
78 

Hydrogen  Hz    .          ... 

Helium  He  
Carbon  C 

Calcium  Ca  

Zinc  Zn  
Cadmium  C'd 

Mercury  Hg  

62 


A   TREATISE   ON   THE   SUN'S   RADIATION 


Hence,  the  following  general  hyperbolic  laws  prevail: 
Bottom  of  the  isothermal  layer,       m  .  ZA  =  —  12000  kilometers 
Top  of  the  isothermal  layer,  m  .  z1  =  +    3600          " 

Depth  of  the  isothermal  layer,  m.(zI—ZA)  =       15600          " 

The  corresponding  depths  for  any  other  gas  can  be  found  on 
the  assumption  that  it  is  monatomic,  but  since  other  molecular 
combinations  may  prevail  above  a  certain  level  within  the 
isothermal  layer,  the  conditions  on  the  sun  become  so  com- 
plicated by  the  association  of  the  atoms  into  molecules  of  different 
configurences  and  valences  that  it  is  necessary  to  reserve  this 
portion  of  the  subject. 

The  Heights  at  which  the  Same  Temperature  Occurs  for  the  Different 
Elements,  by  the  Hyperbolic  Law,  m  z  =  CT. 

An  inspection  of  Fig.  10,  which  gives  the  temperature  lines 
of  the  different  elements,  on  the  same  scales  of  ordinates  and 
abscissas,  suggests  that  the  curves  form  a  family  whose  par- 
ameter is  m,  connected  by  the  law  of  the  equilateral  hyperbola, 
m  z  =  CT,  where  CT  is  a  constant  for  a  selected  temperature  T. 
To  test  this  theorem,  the  heights  z  at  which  the  same  tempera- 
ture T  occurs  for  the  different  elements  m,  was  interpolated  from 
the  original  tables,  and  the  results  are  collected  in  Table  8  for 
certain  temperatures,  T  =  000°,  1000°,  2000°,  .  .  .  12000°.  The 
first  column  contains  the  element  and  its  atomic  weight  m,  the 
second  the  height  z  at  which  the  temperature  T  was  computed 
by  the  thermodynamic  formulas,  the  third  the  product  m  z.  It 
is  easily  seen  that  the  products  tend  toward  a  constant  value 
for  the  same  T,  and  the  mean  is  the  constant  CT,  referred  to 
the  photosphere.  Hence,  we  obtain  the  hyperbolic  constants 
from  which  the  height  may  be  computed  where  the  given  tem- 
perature occurs  in  the  different  elements. 


T  =  0°  occurs  at  the  he  ght 
1000 
2000 
3000 
4000 
5000 
6000 
7000 


45400/m  in  kilometers 
37792 
30246 
24304 
19611 
15422 
11713 
7826 


COMPUTATION  OF  THE  THERMODYNAMIC  TERMS    .   63 

Isothermal  Layer 

8000  occurs  at  the  height  —  14036/m  in  kilometers 

9000  "  "  "  "  -16594  "  " 
10000  "  "  "  "  -18972  "  " 
11000  "  "  "  "  -21223  "  " 
12000  "  "  "  "  -23522  "  " 

The  non-adiabatic  group  can  be  referred  to  the  bottom  of 
the  isothermal  layers  by  adding  12000  to  the  respective  values 
CT,  as  45400  +  12000  =  57400  for  the  temperature  T  =  0°. 
The  first  group  for  T  =  0°  gives  the  height  of  the  vanishing 
plane  of  the  several  elements.  Hydrogen  does  not  exist  in  the 
monatomic  state,  m  =  1.00  above  the  photosphere,  but  only  in 
the  diatomic  form  of  the  molecule  for  m  =  2.00.  It  is,  how- 
ever, very  convenient  to  have  H  =  1.00  fully  computed,  be- 
cause it  is  the  implied  standard  of  reference  for  all  the  chemical 
elements  and  the  entire  scheme  is  more  readily  understood  by 
reference  to  this  H  =  1.00,  as  if  it  actually  existed.  It  will 
be  shown  that  the  molecule  Hz  =  2.00  does  not  exist  below  a 
certain  plane  in  the  isothermal  layer,  but  only  in  the  atomic 
form  Hi  =  1.00.  It  was  necessary  to  use  HI  =  1.00  in  certain 
levels,  and  Hz  =  2.00  in  other  levels,  and  this  constitutes  an 
important  piece  of  evidence  that  at  certain  values  of  P,  p,  R,  T, 
there  is  dissociation  of  H2  into  two  atoms  of  fli,  or  conversely 
that  at  this  level  there  is  association  of  two  atoms  of  HI  into 
one  molecule  Hz.  An  examination  of  the  products  m  z  of  Table  8 
seems  to  indicate  that  the  variations  are  quite  accidental  in 
their  general  arrangement.  It  should  be  remembered  that  the 
data  have  been  obtained  by  the  method  of  trials  applied  in 
succession  to  eight  elements,  so  that  they  are  entirely  in- 
dependent of  one  another.  The  general  process  of  the  compu- 
tations, therefore,  resulted  in  a  family  of  equilateral  hyperbolas, 
which  can  be  readily  developed  for  the  fundamental  theory  of 
the  distribution  of  the  solar  elements.  Since  the  Naperian 
logarithms  are  based  upon  the  law  of  the  equilateral  hyperbola, 
it  follows  that  an  exponential  law  of  decay  or  depletion  must 
be  placed  at  the  foundations  of  solar  gaseous  distributions  in 
height  under  the  force  of  the  gravity  acceleration  there  existing. 
As  already  stated,  it  was  found  that  the  first  four  elements 


64 


A  TREATISE   ON  THE   SUN  S   RADIATION 


TABLE  8 

DISTRIBUTION  OF  THE  TEMPERATURE  HEIGHTS  BY  THE  HYPERBOLIC  LAW, 

mz=  (CONST.)  r 


Temp. 

r=  000° 

T=  1000° 

r  =  2000° 

T  =  3000° 

T  =  4000° 

m 

3 

ms 

z 

mz 

z 

mz 

z 

mz 

z 

mz 

Hi    1.. 

48000 

48000 

42000 

42000 

32900 

32900 

26800 

26800 

21500 

21500 

H^    2.. 

25500 

51000 

19740 

39480 

14710 

29200 

11460 

22920 

8720 

17440 

He    4.. 

11500 

46000 

9500 

38000 

7680 

30720 

6060 

24240 

4720 

18880 

C    12 

4000 

48000 

3280 

39360 

2690 

32280 

2040 

24480 

1660 

19920 

Ca   40.. 

1050 

42000 

880 

35200 

727 

29080 

594 

23760 

472 

18880 

Zn   65.. 

675 

43900 

565 

36725 

483 

31395 

408 

26520 

335 

21775 

Cd   112.. 

380 

42600 

312 

34944 

247 

27664 

196 

21952 

151 

16912 

Hg  198.  . 

210 

41500 

185 

36630 

144 

28512 

120 

23760 

109 

21582 

To  the  Phot 

osphere 

45400 

37792 

30246 

24304 

19611 

12000 

12000 

12000 

12000 

12000 

Bottom  of 

Isother- 

57400 

49792 

42246 

36304 

31611 

mal  layer 

Temp. 

r  =  5000° 

T  =  6000° 

r=  7000° 

m 

z 

mz 

z 

mz 

z 

mz 

The  temperature  of  the  top  of 

the  isothermal  layer  is  7700;   of 

the  bottom  7620;  of  the  photo- 

Hi         I.. 

16600 

16600 

12400 

12400 

8400 

8400 

sphere  7687;  of  the  radiation  layer 

H2          2.. 

6560 

13120 

4800 

9600 

2910       5820 

7655. 

He         4.. 

3670 

14680 

2720 

10080 

1770        7080 

C          12.. 

1500 

18000 

1160 

13920 

800 

9600 

The   constant  *for  the  photo- 

Ca       40.  . 

370 

14800 

277 

11080 

226 

9040 

sphere  +  12000      becomes      the 

Zn       65.  . 

263 

17095 

196 

12740 

131 

8515 

constant  for  the  bottom   of   the 

Cd     112.. 

120 

13440 

93 

10416 

68 

7616 

isothermal  layer. 

Hg     198.. 

79 

15642 

68 

13464 

33 

6534 

To  the  Phot 

osphere 

15422 

11713 

7826 

Bottom  of 

Isother- 

12000 

12000 

12000 

mal  layer 

27422 

23713 

19826 

Temp. 

T  =  8000° 

r  =  9000° 

T  =  10000° 

T=  11000° 

T=  12000° 

m 

z 

mz 

z 

mz 

z 

mz 

z 

mz 

z 

mz 

Hi    1  .  . 

-13000 

-13000 

-15062 

-15062 

-17200 

-17200 

-19300 

-19300 

-21450 

-21450 

#2      2.. 

-  8000 

-16000 

-  9740 

-19580 

-11207 

-22414 

-12630 

-25260 

-14080 

-28160 

He    4.. 

-  3220 

-12880 

-  3772 

-15088 

-  4200 

-16800 

-  4815 

-19260 

-  5345 

-21380 

C    12.. 

-  1267 

-15204 

-  1482 

-17784 

-  1660 

-19820 

-  1827 

-21924 

-  2000 

-24000 

Ca   40.. 

-  366 

-14640 

-  418 

-16720 

-  472 

-18880 

-  523 

-20920 

-  576 

-23040 

Zn   65.. 

-  211 

-13715 

-  245 

-15925 

-  275 

-17875 

-  309 

-20085 

-  342 

-22230 

Cd   112.  . 

-  107 

-11984 

-  130 

-14560 

-  149 

-16680 

-  167 

-18704 

-  189 

-21168 

Hg   198.  . 

-   53 

-10494 

-   75 

-14850 

-   86 

-17028 

-   96 

-19008 

-  107 

-21186 

To  the  Phot 

osphere 

-13490 

-16196 

-18337 

-20558 

-22827 

COMPUTATION  OF  THE  THERMODYNAMIC  TERMS 


65 


pointed  to  the  existence  of  this  elementary  law,  so  that  it  became 
easy  to  locate  the  temperature  curves  of  He,  C,  Zn,  Cd,  -from 
which  fact  the  later  computations  were  greatly  facilitated. 
On  Fig.  14,  the  values  of  CT  have  been  plotted,  showing  that 


GOOOO 


40000 


30000 


20000 


10000 


-10000 


-2000(7 


~30000 


0       2000 
T  Temperature 


4000 


6000 


8000 


10000 


12000 


FIG.  14.     Distribution  of  the  Temperature  Heights  by  the  Hyperbolic  Law, 

mz  =  (Const)  r- 

there,  is  a  convex  curvature  downward  in  the  non-adiabatic 
region,  a  nearly  vertical  line  in  the  isothermal  region,  and  a 
straight  line  in  the  adiabatic  region.  An  auxiliary  adiabatic 
line  has  been  drawn  in  the  non-adiabatic  region,  and  by  resolu- 
tion the  non-adiabatic  curve  can  be  broken  up  into  its  adiabatic 


66  A   TREATISE   ON  THE   SUN'S   RADIATION 

and  isothermal  components.  The  non-adiabatic  branch  ter- 
minates on  the  ordinate  T  =  0°,  but  the  adiabatic  branch  can 
be  extended  as  far  downward  as  the  law  of  gases  prevails. 


The  Constant  Temperature  Near  the  Photosphere  and  the  Variations 
of  Temperature  Outside  of  the  Isothermal  Region 

The  temperatures  of  the  different  gases  were  computed  at 
convenient  intervals,  differing  from  one  gas  to  the  other,  but 
by  interpolation  the  temperatures  for  the  eight  selected  gases 
can  be  found  on  any  level.  Table  6  contains  such  a  compilation 
of  solar  temperatures  from  25,000  kilometers  above  the  plane 
of  the  photosphere  to  —  14000  kilometers  below  that  plane. 
It  is  seen  that  the  table  divides  itself  by  a  broken  line  into 
three  parts,  the  middle  containing  nearly  constant  or  isothermal 
temperatures;  the  upper  part  in  which  the  temperatures  on  the 
same  level  decrease  from  a  maximum  for  HI  =  1.00  to  a  minimum 
or  to  0°  for  the  heavier  elements;  and  the  lower  part  in  which 
the  temperature  on  the  same  level  increases  from  a  minimum 
for  HI  =  1.00  to  a  maximum  for  the  heaviest  element  Hg  — 
198.  Taking  the  mean  values  in  the  isothermal  region,  accord- 
ing to  the  number  of  elements  on  the  left  of  the  broken  line 
of  division,  it  is  seen  that  the  temperature  is  nearly  a  constant 
throughout  this  region,  and  that  the  general  mean  temperature 
of  the  isothermal  region  is  7686.°7.  It  is  noted  that  the  depth 
of  the  isothermal  region  is  great  for  the  light  gases,  and  small 
for  the  heavy  gases,  as  15600  kilometers  for  HI  =  1.00  and 
78  kilometers  for  Hg  =  198,  according  to  the  hyperbolic  laws. 

On  the  other  hand,  outside  of  the  isothermal  region,  on  the 
same  level  there  are  enormous  changes  in  temperature  from 
the  light  to  the  heavy  gases,  as  7696°  to  0°  on  the  1000-kilometer 
level,  or  7651°  to  945470°  on  the  -  10000-kilometer  level.  In 
the  adiabatic  region  the  temperatures  have  been  extended  below 
the  actual  levels  of  the  computations  by  merely  adding  the 
adiabatic  gradients.  These  can  be  studied  for  what  they  may 
be  worth,  but  it  is  thought  that  at  such  high  temperatures, 
pressures,  densities  as  are  implied,  the  simple  gaseous  law, 


COMPUTATION  OF  THE  THERMODYNAMIC  TERMS       67 

P  =  p  R  T,  must   become   greatly  modified   in  the  passing  of 
the  gas  into  its  liquid,  viscous,  or  solid  state. 

It  is  perhaps  superfluous  to  remark  that  in  the  solar  en- 
velope there  is  no  constant  temperature  except  in  the  isothermal 
layers  near  the  photosphere,  and  that  the  popular  idea  of  think- 
ing of  the  sun  as  having  fixed  temperatures  on  the  same  level 
is  erroneous.  By  equation  (6)  it  is  seen  that  temperature  has 
no  dimensions,  such  as  pressure,  density,  and  gas  efficiency 
possess,  but  that  it  merely  represents  a  sort  of  thermal  equilibrium 


-I 
g'J 


T  being  merely  a  ratio,  the  same  value  occurs  at  different  heights 
for  the  different  elements,  or  else  different  values  of  the  ratio 
T  occur  on  the  same  level  for  the  different  elements.  Indeed, 
the  temperature  ratios  are  so  very  complicated  in  all  atmos- 
pheres composed  of  different  gases  as  to  become  intelligible 
only  after  special  analyses.  The  energy  of  radiation  originates 
as  a  special  effect  within  the  isothermal  layers  of  the  different 
gases,  so  that  it  is  proper  to  speak  of  a  constant  temperature  of 
radiation,  but  this  ignores  the  complex  phenomena  that  exist 
in  the  non-adiabatic  and  the  adiabatic  regions.  Similarly,  the 
temperature  of  a  star  as  determined  in  the  spectrum  refers  only 
to  the  source  of  radiation  in  its  isothermal  layer,  and  not  to  the 
other  temperatures  above  or  below  that  region.  It  should  be 
noted  that  on  Fig.  10,  and  in  Table  6,  the  temperature  of  the 
black  body  solar  radiation  is  somewhat  less  than  7687°  Centigrade 
Absolute,  and  it  will  be  shown  to  be  7655°.  There  is  nothing 
characteristic  of  the  solar  radiation  at  the  temperature  5810°,  as 
deduced  from  the  pyrheliometer  data  by  Mr.  Abbot  and-  other  ob- 
servers. Indeed,  it  only  remains  to  determine  the  value  of  the 
coefficient  a  in, 

(92)  /0 

to  obtain  the  solar  constant  of  radiation  at  the  distance  of  the 
earth.  We  shall  show  that  a  at  the  sun's  isothermal  layer  agrees 


68 

with  the  Kurlbaum  coefficient,  and  hence  that  JQ  =  5.85  gr. 
cal./cm.2  min.  before  any  depletions  occur  in  the  solar  and  in 
the  terrestrial  atmospheres. 

The  Distribution  of  the  Pressures 

The  values  of  'P,  the  pressure  in  kilograms  per  meter2,  and 
the  values  of  A,  the  pressure  in  standard  terrestrial  atmospheres, 
have  been  collected  in  Tables  9  and  10  respectively,  for  the  same 
values  of  z,  for  the  eight  elements.  The  original  computations 
contain  many  more  points  between  z  =  1000  and  z  =  25000, 
or  the  vanishing  plane  of  the  gas.  It  is  necessary  to  use  P  in 
all  thermodynamic  discussions,  but  A  is  a  practical  unit  that 
may  be  conveniently  employed  in  descriptive  explanations, 
and  its  meaning  is  easily  understood  in  the  different  levels.  It 
is  to  be  noted  that  P  has  almost  exactly  the  same  value  on  the 
plane  of  the  photosphere  for  each  gas,  the  mean  value  being, 

P  =  616190  kil./m.2, 

A  =  6.0814  standard  atmospheres. 

There  is  no  other  plane  on  which  this  relation  holds,  all  the 
pressures  diminishing  in  horizontal  lines  above  the  photosphere, 
from  HI  to  the  heavy  gases,  and  increasing  below  the  photo- 
sphere from  HI  to  the  heavy  gases.  In  the  adiabatic  region, 
below  the  photosphere,  these  computations  can  be  extended  to 
enormous  pressures  corresponding  with  the  great  temperatures. 
On  the  other  hand,  in  vertical  directions,  the  hyperbolic 
law  m  z  =  (Const)p  for  a  selected  pressure  holds  true  as  shown 
by  Table  11.  The  height  of  the  vanishing  plane  is  seen  under 
the  first  section  for  A  =  0,  where  the  product  approximates  the 
mean  value  m  z  =  45002,  and  similarly  in  the  following  sections. 


COMPUTATION  OF  THE  THEEMODYNAMIC  TERMS 


69 


TABLE  9 

THE  PRESSURE  OF  THE  DIFFERENT  ELEMENTS  ON  THE  SAME  LEVELS 
(Kilograms  /meter2),  P  (M.  K.  S.) 


z 

Kilom- 
eters 

Hi 
1.00 

H2 

2.00 

He 
4 

C 
12 

Ca 
40 

Zn 
65 

Cd 
112 

Hg 
198 

Means 

25000 
20000 
15000 
10000 
5000 

1000 
900 
800 
700 
600 
500 
400 
300 
200 
100 
90 
80 
70 
60 
50 
40 
30 
20 
10 
Photos. 
-10 
-20 
-30 
-40 
-50 
60 

2648.2 
12877 
44710 
120450 
276700 

515610 
523640 
531780 
540000 
548440 
556960 
565630 
574430 
583350 
592410 
593350 
594250 
595170 
596090 
597010 
597930 
598860 
599780 
600700 
601630 
602570 
603500 
604450 
605390 
606330 
607270 
608210 
609160 
610110 
611060 
620670 
630430 
640360 
650430 
660670 
671060 
681620 
692340 
703230 
821980 
958160 
1116920 
1305550 
1526000 
1778800 
2073500 
2426700 
2839900 
3348700 
3858900 
n 
-9000 
2426700 

2  .  727x10 
0.03778 
110.52 
8165.6 
108265 

440880 
454850 
468670 
483210 
498220 
513688 
529640 
546110 
563090 
580600 
592390 
584170 
585960 
587760 
589560 
591370 
593190 
595000 
596830 
598657 
600550 
602460 
604390 
606300 
608230 
610170 
612110 
614060 
616010 
617970 
637910 
658490 
679730 
701667 
723480 
746020 
769250 
793220 
817920 
1115480 
1522400 
2078830 
2839900 
3874000 
5287000 
7158670 
9555800 
12498900 
20247000 
30917100 

-4500 
2459365 

Top  of 

the  Inner 

Corona 

4 

0.24958 
10770.3 

328086 
348870 
371020 
394560 
419610 
446250 
474760 
505060 
537290 
571590 
575100 
578670 
582260 
585890 
589530 
593190 
596870 
600580 
604310 
608063 
611790 
615510 
619290 
623060 
626870 
630700 
634560 
638430 
642330 
646250 
686820 
729930 
775750 
824440 
877240 
933400 
993160 
1056760 
1124390 
2082760 
3873360 
6776430 
10718300 
15786300 
22074500 

Top  of 

Chromo 

sphere 

92408 
114260 
141280 
172020 
209435 
252660 
304807 
367570 
443250 
534560 
544300 
554210 
564320 
574000 
585070 
595740 
606600 
617620 
628860 
640370 
652700 
665240 
678050 
691080 
704380 
717930 
731750 
745820 
760170 
774780 
931240 
1119300 
1345370 
1622400 
1956450 
2357200 
2839900 
342350Q 
4127100 
19705500 
52206000 
105061000 

-5 

7.755x10 
0.46894 
26.881 
362.57 
2350.4 
9785.8 
30407 
75783 
148280 
287830 
306410 
326180 
347230 
369640 
393510 
418850 
445830 
474540 
505110 
537620 
571630 
607760 
646200 
687070 
730520 
777570 
827660 
880960 
937720 
998070 
2064900 
3871580 
6955330 
11488400 
17435200 
24918300 
34050000 
44937800 
57678750 
306112700 

0.771869 
83.238 
2110.5 
16411 
73052 
231670 
256110 
283120 
312970 
345990 
382510 
422470 
466620 
515380 
569230 
628814 
695150 
768500 
849580 
939200 
1038290 
1147800 
1268900 
1402800 
1550800 
1714440 
4700900 
10972600 

Reversi 

ng  Layer 

3.7724 
4185.1 
110313 
140109 
169904 
209405 
248906 
301903 
354900 
428378 
501856 
606612 
711367 
856834 
1002300 
1211515 
1420730 
1713140 
2005550 
2422725 
2839900 
3425725 
4011550 
15310000 

3.150x10 
11380 
19967 
32999 
51958 
78746 
115760 
165960 
233350 
323060 
442456 
603000 
816820 
1121320 
1526210 
2090450 
2839900 
3853640 
5144000 
6682670 
8468400 
10514000 





616190 



2593212 

70 

80 

90 

-100 
-200 
-300 
-400 
-500 
-600 
-700 
-800 
-900 
-1000 
-2000 
-3000 
-4000 
-5000 
-6000 
-7000 
-8000 
-9000 
-10000 
-12000 
-14000 
Radiatio 
layer  ZR 
Pressure 

-2250 
2530410 

-750 
2598550 

-225 
2516570 

-140 
2909024 

-80 
2839900 

-45 
2465175 

2593212 

70 


A   TREATISE  ON  THE   SUN  S  RADIATION 


i 

i 

co 

6«00 

&:£ 

1 

13 

rH    t-   00  CM   "tf 

CO   CO   rH   CO 

CO 
IO 

os 

eo 

rH 

rH  CM  CO  ^ 

IO 

S3 

a 

Reversing  Layer 

3.723X10-6 
.04130 
1  .  0887 

CO  CO  O  tf  OS 
iH  r-i  CM  CM  CM 

IO   rH    OS   00 

co  •*  **"  10' 

t-' 

,5s 

i  1      ,1 

fcH               O 

S          Q 

~    £ 

bs^ 

06             'CM 

t-   CO   rH   OS   rH 

t-  •*  OS  -<a<  IO 
CM  OS  CO  rH  t- 

10    t-    0    T*    t- 

CM  CM  CO  CO  CO 

IO  -*  OS  CO 

sill 

•^  r)I  10  10 

CO 

f  T  r 

a)                         ooo 

g                                              rH    rH    rH    OS    t- 

oo  os  eo 

SJ  "* 

O                                    -^  00  CO  CO  CO 
rj                                    kO  CM  kO  O  CM 

co  co  co  o  o 

V                                    t>  TJ  CM      ' 

9 

co  o  t-  eo  o 
So  •<*  co  Tf 
CO  t>  rj<  00 

'i-i  CM 

CM   rH   CM   IO   OO 

0  <N  ^  CO  00 

co  eo'  eo  co'  eo 

eo  os  oo  oo 

rH   CO   CO   05 

iO 

r  x   CM 

5                cM£3£°. 

l^                                          rH   CM   OS   CM   CO 
O                             i            OS  rH   CO   rH   O 

CO   CM   t-   CO   rH 

eo  oo  t-  •**  oo 

OS  O  CM  t-  kO 
^J<  O  CO  CO  CM 

IO  IO  CO  OS  Tj* 

eo  •«*  10  co  t- 

llll 

1 

O.                                         O  rH   rH   rH   CM 

CM  CO  CO  •«*  IO 

10  10  10  10  10 

10  10  CO  CO 

CO 

S* 

I 

O 

I/N   CO           ^  rH   t~   rH   CO 
CO  <O           t-   CO   rH   •<*   rH 

co  o       eo  •>*  co  os  -* 

•^   rH           CM   rl<   CO   00   rH 

CM   CO   CO  00  rH 

CO  t*  t*  t*  00 

nil 

i 

CM  o      co  eo  co  eo  •«* 

T*    •<*    T*    10    10 

1010101010 

10  10  10  10 

CO 

'k 

O5NOOOCO         lOOOCMCOrH 

cct-ooo       eo^cot-os 

Nil! 

CO  Tj<  rH  OS  CO 

^SS§2 
t-  t-  t-  00  00 

SS8§ 

eo  10  t-  os 

00  OO  OO  00 

i 

10  10  10  10  10 

10  10  10  10  IO 

1O  1O  IO  IO 

kO 

*8 

co 

CO  CM  O 
COrHCOOOrH           OOOOOOOSCM 
OO^rHtS           OrHCMCO^ 

Hill 

O5  OS  O  O  rH 
IO  •<*  T}I  CO  CM 
IO  CO  t-  00  OS 
00  OO  OO  OO  00 

rH   O   OS   00 

os  os  os  os 

t- 

OOOrHCM         1O1O1O1O1O 

IO  IO  IO  IO  IO 

10  10  10  10  10 

IO  IO  IO  1O 

10 

<D 

.... 

o 

s 

CM   CM  rH   rH 

§1111 

d  o  o  o'  o 

OS  00  t-  CO  10 

ssss 

e 

CH 

COMPUTATION  OF  THE  THERMODYNAMIC  TERMS 


71 


TH          OS  O 


CO  O  00  O         CO  t-  10 

oo  o'  TH  Tji       eo' 

t-  kO  00  Tj<  

00  IO  CO  CM    CM  CO  IO  00  CO    Oi  CO  CM  00 

co't-'odos       OT-ICMCOIO       «Deoooeo 

SSt2o       c^Sooo^S       §0010       ^eooo^e,  coeo_ 

n<  os  t-  oo       o  t-  eo  os  10       10  oo  TH  ti<       oo  t-  co  10  ,H  eo-^ 

eo  os  co  t-       cMcoi-Heooi       ooWCMeo       cooosoio  CMiH 

10  10'  eo  CD   t-'  t>  oo  oo  os   os  o  oo  oo   co  CM  jo  eo  eo  g  ^ 

TH  1-1  CM  eo  •*  10  o 

eo 

t^.^4^00         iOt^-CO^CM          XOOO 

OSi-liOtH         Oi-HiOi-HO         ^H0t>00         CMOSrjtOOOO  N 

COCOOOiH         ^*t~O^fOO         CMOST^t—         i— (  O  <^  CM  00  COOOCM 

Tfiocooo       OSOCMCO^*       eor-iocM       oeoe^ot-  t-Tfuj 
eoeoeoeo       eot-t»i>t-       t-osi-ieo 

OOOOCO         OSCDOSOSTl<         i-l-^OCM         OOOOCMO 

oo-^coos       eo-^CMOOs       oooo-^eo       ecc-NCMO  ioocoosr-i 

cot— THT}<       OOCMCDOCO       t—  t-  o  10       eocOi-HOco  ooi^CMt— oo       oco 

O  O  i— *  iH          i-HCMC^COCO         CO  t—  <N  eo         iHCOWOO^  OlflCMOOC—         0000 

eo'  eo'  co  CD'   co  co  eo'  eo'  CD   eo"  co  t-  t-   oo  oo  os  os  o  TH  o  oo  eo  10   us  t- 

1—1  i-HCMCOCOO  kOi-H 

i-H  iH   CM 


SCO  00  O  U5         •Tf 
OCMCMOO         CMOSlOt>. 
COOSN         t>OCM^H 
OOOift 


iHOOSOS         OSOr-lTjtf-         OOO 

io  10  10  io   eo  co  eo  eo  eo   eo  eo'  co  CD'   eo  t-'  t-'  t-'  t-'   oo  TH  jo  o  oo   oo  CM  O  TJ j  eo 


rjlO 

eOOO 
CMrH 


10  10       10  10  eo  eo  eo 


r-l   CO   O*    OS    00 

eoeoeo       eoeoeocoeo       CDOOOST-ICM       lot-oeooo 


osoeot-c<i       ocMco 

i-HNNCMCO         ^i-HiO 
rJ<iOCOt-00         OSi-HTl< 


0000 
TH  (N  CO  ^ 

I  I  1  1 


»oeot-ooos       oooo 

I      I      I      I      I  r-t  CM  CO  ^ 

IIIM  I      I      I      I 


cot-ooos 


T    i  i 


72 


A   TREATISE   ON   THE   SUN'S   RADIATION 


The  mean  pressure  for  all  elements  at  the  levels  where  black 
radiation  originates  is  2593212  kilograms  per  square  meter,  or 
kinetic  energy  per  cubic  meter. 

TABLE  11 

THE  DISTRIBUTION  OF  THE  HEIGHTS  FOR  PRESSURE  BY  THE  HYPERBOLIC 
LAW,  mz  =  (CONST.)P 


logP 

_ 

5.0057 

5.3067 

5.4828 

5  .  6078 

A 

0 

1 

2 

3 

4 

m 

z 

mz 

I 

mz 

2 

mz 

z 

mz 

Z 

mz 

Hi     I... 

48000 

48000 

10910 

10910 

6940 

6940 

4400 

4400 

2550 

2550 

Hi    2... 

25000 

50000 

5660 

11320 

3340 

6680 

2700 

5400 

1770 

3540 

He    4... 

11000 

44000 

3210 

12840 

1750 

7000 

1120 

4480 

660 

2640 

C    12... 

4000 

48000 

957 

12440 

616 

8008 

402 

5226 

247 

3211 

Ca    40.  .  . 

1050 

42000 

253 

10120 

154 

6160 

91 

3600 

50 

2000 

Zn    65... 

675 

43875 

174 

11375 

112 

7280 

73 

4745 

44 

2860 

Cd   112... 

380 

42560 

103 

11536 

71 

7952 

49 

5488 

32 

3584 

Hg   198... 

210 

41580 

63 

12474 

34 

6732 

22 

4356 

13 

2574 

Means  

45002 

11627 

7094 

4712 

2870 

logP 

5.7047 

5.7839 

5.8508 

5.9088 

5.9600 

A 

5 

6 

7 

8 

9 

m 

2 

mz 

Z 

mz 

z 

mz 

Z 

mz 

z 

mz 

Hi          1... 

1180 

1180 

-80 

-80 

-1070 

-1070 

-1810 

-1810 

-2680 

-26SO 

Hz          2... 

540 

1080 

-60 

-120 

-540 

-1080 

-980 

-1960 

-1345 

-2690 

He         4... 

250 

1000 

0 

0 

-254 

-1016 

-473 

-1892 

-663 

-2652 

C          12... 

127 

1524 

+  31 

+  372 

-55 

-660 

-127 

-1651 

-189 

-2457 

Ca        40... 

29 

1160 

-10 

-400 

-26 

-1040 

-47 

-1880 

-66 

-2640 

Zn        65... 

22 

1430 

+  3 

+  195 

-12 

-780 

-25 

-1625 

-37 

-2405 

Cd      112... 

12 

1344 

+  1 

+  112 

-7 

-784 

-16 

-1792 

-23 

-2576 

Hg      198.  .  . 

6 

1238 

0 

0 

-6 

—1188 

-10 

-1980 

-13 

-2574 

Means  

1188 

10 

-952 

-1824 

-2582 

.  logP 

6.0057 

6.3067 

6.4828 

6.6078 

6.7047 

A 

10 

20 

30 

40 

.  50 

m 

z 

mz 

Z 

mz 

z 

mz 

Z 

mz 

Z 

mz 

Hi     1... 

-3350 

-3350 

-7870 

-7870 

-10450 

-10450 

-12300 

-12300 

-13165 

-13165 

Hz    2... 

-1690 

-3380 

-3930 

-7960 

-5220 

-10440 

-6150 

-12300 

-6870 

-13740 

He    4... 

-833 

-3332 

-1963 

-7852 

-2610 

-10440 

-3078 

-12312 

-3435 

-13400 

C    12... 

-246 

-3198 

-620 

-8060 

-837 

-10881 

-990 

-12870 

-1112 

-13344 

Ca   40... 

-85 

-3400 

-198 

-7920 

-261 

-10440 

-307 

-12280 

-340 

-13600 

Zn   65... 

-48 

-3120 

-117 

-7605 

-157 

-10205 

-185 

-12025 

-208 

-13520 

Cd   112... 

-30 

-3360 

-69 

-7728 

-94 

-10528 

-111 

-12432 

-122 

-13664 

Hg  198.  .  . 

-17 

-3366 

-39 

-7722 

-52 

-10298 

-62 

-12276 

-69 

-13662 

Means  

-3313 

-7840 

-10460 

-12349 

-13412 

COMPUTATION  OF  THE  THERMODYNAMIC  TERMS 


73 


log  P  is  the  logarithm  of  the  pressure  P  in  kilos./meter2. 

A  =  the  pressure  in  terrestrial  atmospheres. 

Unit  A  =  101323.5  kilos./meter2  [5.00571]. 

logP  -  5.00571  =  log.  A. 

A  was  selected  at  the  numbers  0  .  1  .  2  .  3  ...  as  given. 

From  the  computed  pressures,  at  arbitrary  heights  z,  the 
height  corresponding  to  A  was  interpolated,  and  the  product 
m  z  for  each  A  is  a  constant,  the  variations  seen  in  the  table 
depending  upon  the  inaccuracy  of  T  determined  by  the  method 
of  trials.  By  working  backward  from  these  constants  for  T 
and  P  more  accurate  computations  for  the  entire  thermodynamic 
system  can  be  made. 


A  = 


0  log  P               occurs  at  the  height      45002/m  in  kilometers 

1 

=  5.0057 

11627 

2 

•'..     5.3067 

7094 

3 

5.4828 

4712 

4 

5.6078 

2870 

5 

5.7047 

1238 

6 

5.7839 

10 

7 

'    '<    5.8508 

—     952 

8 

5.9088 

-  1824 

9 

5.9600 

-  2582 

10 

6.0057 

-  3313 

20 

6.3067 

-  7840 

30 

6.4828 

-10460 

40 

6.6078 

-12349 

50 

6.7047 

-13512 

In  the  region  of  the  adiabatic  layers,  as  10  to  50  A ,  it  may 
be  stated  that  the  differences  in  the  constants  decrease  in  value, 
4527,  2620,  1889,  1163,  so  that  finally  the  pressure  approaches 
asymptotically  to  a  maximum.  The  same  is  true  for  each  gas 
in  the  differences  (log  PQ  —  log  PI)  in  succession,  the  maximum 
for  each  gas  being  the  end  of  the  gaseous  stage,  or  entrance  into 
the  viscous  or  quasisolid  stage,  the  change  being  very  gradual 
on  approaching  the  asymptote.  Further  computations  are  de- 
sirable to  bring  out  the  maximum  for  each  gas  and  the  depth  at 
which  it  occurs,  these  depths  being  very  different  for  the  several 
gases.  The  entire  independence  of  the  distribution  of  the  gases 
in  the  family  of  curves  conforms  to  the  fundamental 


74 


A   TREATISE   ON  THE   SUN'S   RADIATION 


q<?  = 


(93)  Laws  of  Dalton,      P  =  Pl  +  P2  +    .  .  .  P2. 

(94)  Clausius,    \  m  q2  =  \  mi  qi*  =  J 

(95)  Avogadro,  N  =  constant  for  (P  .  V  .  T)  constant. 
Fig.  15  shows  for  the  pressure  (constants)P  their  change  in 

height  in  passing  through  the  non-adiabatic,  isothermal,  and 


50000 


.40000 


30000 


20000 


0  10  20  30  40 

FIG.  15.     The  Distribution  of  the  Pressure  Heights  by  the  Hyperbolic  Law. 

mz  =  (Const.)  p. 

adiabatic  layers,  which  is  to  be  continued  to  the  asymptotic 
maximum. 

The  Distribution  of  the  Densities 

Collecting  the  data  for  the  densities  in  Table  12  in  the  same 
way  as  for  the  temperatures  and  pressures,  we  note  that  there 
are  two  important  characteristics: 

(l)  On  the  plane  of  the  photosphere  the  densities  are  re- 
lated by  the  hyperbolic  law, 

0.000629  X  m  =  p,  or 
p/m  =  0.000629. 


COMPUTATION  OF  THE  THERMODYNAMIC  TERMS       75 

On  no  other  line,  horizontal  or  vertical,  does  this  law  seem  to 
apply. 

(2)  In  the  vertical  columns  the  densities  decrease  by  the 
curves  of  Fig.  12.  On  the  horizontal  lines  there  is  a  maximum, 
that  occurs  just  below  the  height  at  which  the  next  heavier 
element  begins  to  diminish  rapidly  in  its  density.  Below  the 
photosphere  there  is  steady  increase  in  the  density  along  the  line 
of  a  given  level. 

The  density  of  the  earth's  atmosphere  at  the  sea  level  is 
Po  =  1.29305  kil./m.3  The  density  of 

Hydrogen  on  the  photosphere  is  .  p0  (sea-level-air). 

l.Zlo.4: 


Calcium 


54.66 


Cadmium  "     "  "  "  — 


Mercury 


16.575 
1 


10.286 

The  extreme  rarefaction  of  the  solar  gases  above  the  photo- 
sphere, where  the  spectroscopic  observations  are  made,  can  be 
readily  appreciated. 

In  the  flash  spectrum,  where  the  Fraunhofer  lines  originate,  as- 
sumed here  at  500  kilometers,  these  ratios  become, 

Hydrogen,  1/1358.5  p0  (sea-level-air). 

Helium,       I/  611.2 

Carbon,       I/  286.6  maximum. 

Calcium,     I/  604.7 

Zinc,  1/5037.8 

Cadmium,    

Mercury,      

At  the  top  of  the  chromosphere,  5000  kilometers,  hydrogen 
is  1/41102  po  sea-level-air.  Similarly,  other  ratios  can  be  com- 
puted from  the  table.  At  the  depths  to  which  the  computations 
are  here  quoted  only  calcium  at  between  —  2000  and  —  3000 
kilometers  reaches  the  air  density.  The  densities  near  the  disk 
of  the  sun  are,  therefore,  very  small. 


76 


A   TREATISE   ON   THE   SUN  S   RADIATION 


c/) 


W 


<!      W 

H 


I 

.  000629  Xm 

53 

T 

0 

1   XCD 

6        ES 

05   CO   00   i-H   OS 
CO   OS   t*   CO   OS 

O 

sSil 

|S»S 

i 

3      «  • 

.s    T 

2     OT)< 

SS 

SJ 

1 

>  I  XS| 

CD  CO  (M  OS  i-H 
CO  CO  CO  5<  5§ 

co  CD  co  m 

O  CO  OO  CD 

•^r  o  co  co 

I-H  t-  co  o 
m  m  CD  t- 

t- 

0 

oo 

CO 

CO  O  O 

0 

o 

1 

in 

2 

T 

T 

u 

0 

O  OS  Tt«  0  0 
TH  0  t-  0  CD 

1-1  m  -^  co  os 

OO  O  CO  TP  CO 

co  co  oo  m 

UO 

e  in 

*o 

1    X 

X  oo  co  oo  -^ 
co  t-  m  IH 

•^   CD   t-   OS  i-H 

CO  S  t^  0? 

N 

NJ  CO 

o. 

oo 

m  1-1  ^  i-i  co 

00   0           TH   CO 

CO  CO  CO  CO  CO 

CO  CO  CO  CO 

H 

*  ' 

T  T  T 

«  0 

c 

o  o  o  o  co 
^       w  G} 

00  lO  ^  O 

CO  t-  <M  0  O 

oo  I-H  01  co  co 

CO  CO  O  CO  CO 
TH  CO  CO  O5  CO 
CO  -^  t-  O  CD 

Illli 

CD  Tf  in  oo 

0  3  2  CO 

i 

a; 

OS^NO 

0 

o 

°. 

° 

G                ^r-^ 

eg  m  co 

^CO 

1                          • 

eg  CM  co  co  •<* 

""*   OO   S   CO   OO 

eo  m  t-  m  co 

t-  CO  CD  t-  OS 

co  1-1  os  t-  m 

rH   CO   CO   CO   T* 
t>   t-   t>   t-   t> 

o  t>  co  oo 
co  m  o  m 
•^  co  TH  os 
m  co  t~  t- 

S 

oo 

* 

H 

• 

85 

T  T 

0  0 

os  i-l  t-  o  os 

CD  t~  OO  O  r)< 

CD  00  OS  CO  Tj< 

t-  o  co  oo 

^1 

*" 

I  XX 

os  in 

TH   CD 

oo  m  co  m  oo 
in  eg  os  CD  co 
t-  oo  oo  os  o 

pili 

CO  CO  i-l  i-t  O 
CD  C~  OO  OS  O 

os  os  oo  t~ 
m  in  m  m 

CO  CO  CO  CO 

.  002547 

~§ 

1      1      1     CO 

0  O  0  0  CD 

00  CO  t-  CO  -^ 
CO   IN   i-l   i-l   iH 

m  c~  os  TH  co 
oo  oo  oo  os  os 

OO   1>   t-   i-H   Tf 

rH  CO  CO  CO  00 

in  t-  os  I-H  co 
os  os  os  o  o 

t>  o  co  m  oo 

o  co  m  t-  os 

TH     ^J<     t-     0 

llli 

09 

*> 

t-  in  -*  o  o 
o  o  co  o  o 

o 

0 

§ 

o 

o 

o 

0 

O 

9 

t-  00  CO 

gg 

n<  o  I-H  TH  o 

•«*   CO   CO   i-H   OS 

co  co  co  •<*  t- 

0                TH     CO     T* 

oo  c<i  CD  o  m 

CD  (N  t-  CO  OO 

Tt<  oj  m  I-H  co 

m  I-H  CD  co  oo 

t-  oo  oo  os  os 

^t  o  m  TH 

O   »H    TH   CO 

CO  CO  CD  CD 

1 

0 

1 

° 

| 

' 

' 

.     .     . 

: 

£ 

4-> 

O 

o  o  o  O  O 
o  o  o  o  o 

o  o  o  e>  o 
o  o  o  o  o 

§§§§§ 

O  O  O  O  O 
OS  00  t-  CO  10 

3iSS 

0 

w 

o  o  o  o  o 
10  O'  in  o  in 

CO  CO  TH  iH 

O  OS  00  t-  CD 

m  TI<  co  co  r-t 

S 

COMPUTATION  OF  THE  THERMODYNAMIC  TERMS       77 


O  IO  Tf  Tf  CO  Tf  OS  CD  CO 

TfTfO  iO  »O  OS  CO  IO  CD 

COOJIO  OOCOTfCOCO  OO 

00    rH    CO  rH   00   IO   CO   rH  O> 

rHCOCO  eo  co  Tf  10  co  co 


eo  rH  »o  co  O  oo  eo  t~  loeoo 

OOTfrH  O  CO  CO  O  10  CO  OS  Tf 

IO   CO   00  rH    IO   rH    O   OO  O   rH   Tf 

OSOrH  COTfCOt^OS  COO>CO 

rHrH  rHrHrHrHrH  CO^fOO 


li 


III 

O  •«*  00 
«0  CD  <D 


IO  Tf  Tf  Tf 
Tf  IO  CO  t- 

CO  CO  CO  CO 


«5  -I  t>  CD 

CO    CO    Tf    ft 

-^  lO  CD  OO 
00  O5  O  i-H 
N  (M  CO  CO 


CO  IO  CD     rH  Tf 

co  oo  co       o>  co 
co  co  eo      co  o 


00  O  CD  10 
•^f  C7>  CO  00  CO 
i-l  rH  CO  CO  CO 


Ssl 


O  OJOOOU5  OOCOOOCDCO 

co  rHcocoo  cooeocoio 

t-  OOOOOCO  t-Tff^CO'O 

00  OOOSrHCO  COlOt^OSrH 


-f  O  O 
O>  CO  CO 
kO  »O  i-H 
TH  O  00 


CD  ^O  IO 
IO  CO  t-  i 
IO  U5  IO 
CO  CO  CO 


rH  Tf  CO  00  Tf  rH  . 

Tf  eo  co  co  co  co 

O5  O  rH  CO  CO  Tf 

IO  CD  CO  CO  CO  CO 

CO  CO  CO  CO  CO  CO 


CO  CO  O  O>  l- 
t-  CO  Tf  OO  OO 

IO  t>  OS  rH  Tf 


OS     O  rH  CO 

i  co   eo  eo  eo 


eo  eo   eo  w  co 


t-  rH  10  o»  co  oo  co 

Sco  oo  o  co  10  oo 

CO   CO  t-  t-   t-  t- 

o  o  o  o  o  o  o 


VOCOt-CO  rHCOrHTfTf  OrHOrH 

lOOlOrH  OOTfrHOOCO  lOCOrHN 

OOrHCOCO  OOrHTfCOOS  CO^OCOt- 

OrHrHrH  rHCOCOCOCO  COCDOU5 


§  § 


Tf  Oi  O  CO  CO 

10  co  t-  o  o 

05    rH    CD    CO  O 

Tf  CO  t-  04  CO 


lOrH  t-eOO>lOi-l  t>t-OOO» 

Tfio  lococot-oo  ooTfoco 

coco  cococococo  coTfioio 

COCO  COCOCOCDCO  COCDCOCO 


COTfOJIO 

C-JO>O»O 
rncocoo 


CDOTf  OOOlOrHrH 

U3COOO         TfcOCDt-t- 

S§2    SS2SS8     *K«* 
I  "     8"-| 


§§ 


rHCOCOTf  lOCDt-OOOS 

Mil        I    I    II    I 


I  I   I   I   I      I   I   I  I 


llll 

^      t-      00   rH 


iii 

•    •  o 

3 

ko  oo    3 

S!5  « 
gj  .2 

8.8  I 


1 

§§  s 


I 


78  A   TREATISE   ON  THE   SUN?S   RADIATION 

The  Gas  Efficiency  of  the  Different  Elements 
The  gas  efficiency  R  is  the  velocity-square  per  degree  of 
temperature,  and  hence  is  the  kinetic  energy  of  the  gas  mole- 
cules in  their  movements  of  translation,  oscillation,  and  collision, 
corresponding  to  a  change  of  one  degree  of  temperature.  This 
is  the  thermodynamic  coefficient  and  does  not  refer  to  the 
electromagnetic  oscillations  within  the  molecules  and  atoms,  or 
the  forces  concerned  with  association,  dissociation,  and  ionization. 
The  distribution  of  R  is  given  in  Table  13  for  the  same  co- 
ordinates. We  find  that  there  are  two  places  in  which  the 
hyperbolic  law  applies.  Along  the  photosphere, 

R  =  127702/w,  or  R  m  =  127702, 

and  along  the  line  dividing  the  adiabatic  layers  from  the  iso- 
thermal layers,  the  relation  is 

Ra  =  281808/w,  or  Ra  m  =  281808, 

that  is,  approximately  along  the  broken  line,  though  the  transi- 
tion is  not  so  abrupt  in  the  computations.  The  relations  for  R 
are  shown  clearly  on  Fig.  13,  with  a  very  large  range  for  hydro- 
gen, contracting  to  a  small  range  for  mercury.  Hence,  one 
degree  of  temperature  signifies  a  large  change  in  the  velocity- 
square  of  the  hydrogen  molecule,  and  only  a  small  change  in 
the  kinetic  energy  of  the  heavy  mercury  molecule,  in  fact,  in  ac- 
cordance with  Clausius'  Law,  that  kinetic  energies  are  constant 
for  the  different  elements.  The  secondary  deflection  in  the 
curves  for  R  occurs  at  the  height  of  the  greatest  gradient  in  the 
temperatures,  and  is  the  compensation  for  it.  In  the  lower 
levels  of  each  gas  R  becomes  a  vertical  line,  and  these  lines  are 
connected  together  by  the  constant  Ra  m  =  281808.  In  the 
case  of  calcium  the  values  in  the  adiabatic  region  seem  to 
require  a  slight  readjustment  by  a  second  computation.  This 
example  is  left  in  order  to  illustrate  how  by  another  trial  such 
divergences  can  be  removed,  that  is,  by  selecting  somewhat 
lower  values  of  the  temperature  T  in  this  case. 

At  the  levels  of  the  source  of  the  radiation  layers  the  value 


COMPUTATION  OF  THE  THERMODYNAMIC  TERMS       79 

The  Cause  of  the  Sharp  Limb  of  the  Sun 

The  cause  of  the  sharp  visible  edge  of  the  sun,  that  is,  the 
configuration  of  the  disk  in  the  midst  of  a  gaseous  envelope,  has 
been  a  problem  of  much  difficulty  to  explain.  Perhaps  the 
favorite  theory  is  that  this  layer  is  composed  of  a  cloudy  con- 
densation, as  evidenced  by  the  granulation,  the  appearance  of 
the  spots,  faculae,  and  similar  forms.  Another  view  is  that  of 
Schmidt,  who  advocates  the  application  of  circular  refraction, 
by  which  a  ray  of  light  originating  at  great  depths  in  an  at- 
mosphere of  densities  diminishing  from  the  center  outward,  will 
at  a  proper  height  pursue  a  circular  path.  Visible  surface 
phenomena  may  in  this  way  actually  exist  at  much  lower  depths 
than  the  surface  of  the  disk,  so  that  sun-spots  and  the  other 
phenomena  represent  deep-seated  optical  effects.  Julius  has 
applied  the  principles  of  anomalous  dispersion,  or  irregular 
refraction,  among  neighboring  masses  which  differ  greatly  in 
density  from  one  another,  to  account  for  the  prominences,  the 
Fraunhofer  lines,  as  absorption  lines  enveloped  in  dispersion 
bands,  reversals  of  lines  in  the  chromosphere,  some  of  the  sun- 
spot  appearances,  and  similar  effects.  Abbot  intimates  that  the 
effect  of  scattering  upon  the  outgoing  solar  radiation  is  such  as 
to  "prevent  us  from  seeing  toward  the  center  of  the  sun,  when 
looking  directly  at  the  middle  of  the  solar  disk,  to  more  than 
5000  miles  (8047)  kilometers,  below  the  reversing  layer."  "At 
the  edge  of  the  disk,  owing  to  the  oblique  line  of  sight,  gaseous 
scattering  will  probably  extinguish  almost  all  yellow  light  start- 
ing from  more  than  500  miles  below  the  chromosphere,  while 
an  even  less  thickness  suffices  for  blue  or  violet  light."  It  may 
be  remarked  that  all  these  hypothetical  suggestions  make  of 
the  sun's  disk  an  optical  effect  of  some  sort,  rather  than  a  material 
distribution  under  the  thermodynamic  laws  prevailing  in  the 
solar  envelope. 

An  examination  of  Tables  6  for  T,  9  for  P,  12  for  p,  and 
the  corresponding  figures  10  for  T,ll  for  A  or  P,  12  for  p,  makes 
it  evident  that  the  sharp  edge  of  the  solar  disk  is  due  to  the 
rapid  diminution  in  height  of  the  heavier  gases  under  the  solar 


80 


A   TREATISE   ON  THE   SUN  S   RADIATION 


rH  00  10 

a   CM 


i   as 


S  S  S 

t-  00  OS 


U5  OS  N         O  CO  CO  <O 

OS  OS  O         O  iH  <0  (D 


s  s 


i-l  CM  i-i 

t-  co  «o 

<N  CO  •* 


t- •*  t- t- 10      in  T-I  to  o 


Tf    Tj<    00 

i-(  00  10 

t-  t-  oo 


CM  CM  N 


§0 


00  US  00  CD  <O 

O5  00  ifi  t- 

N  «5  00 


iH  N  CM  iH  00         CO  T]|  CO  OS         XO         «O  T)*  Ol  00 


t-  OS  t-  t-  CO         «O  O  O  CM  «0         N  t- 


«O  OS  M<  00  CO 
t-  •<*  CM  OS  t- 

eooo-rf'THoo       ooosooi-i 

'      OOOSOS         OSOSOOO 


CJ  T-I  d 

CO  0  0 


•*  T)< 


00  OS  O 

?o  t-  oo 


O  -^  CO  iH  00 

•«*  eo  Tf  t-  1-1 

CO  O  t-  Tj<  W 


T)<  W 


00  00 


00  CO  N  t-  00 


CMTflO         «o«O<O<O<O         t-t-t- 


oo       os  : 

T-!C<iC>l         CMCOeOCOCO         COCO 


§    g|S 


«Ot-N«O         O 


«o  t-  t- 

NCMN 


I 


§  §§§§ 

O         OOSOOt" 


§§§§ 


oooo      o      oooo 

•fCOCMiH  .         i-INCO** 

8      I   I   I   I 

o 

JS 

a* 


COMPUTATION  OF  THE  THERMODYNAMIC  TERMS 


81 


OS  t- 

°  "-• 


N  10  c>  10  to  N 

O  00  O  t-  00  •<* 

t-  CO  t-  t>  OO  O 

eo  Tji  10  CD  t>  o> 

N  N  M  <N  (N  N 


CO  **     IO  CO  CO         t-OlO 


00  l> 
10  C<« 


•^  00  »-^  «O  ^H 
«O  •*  5O  i-H  O 
Oi  O  i-H  N  CO 


03  CO  CO  CO  CO 


>-!  00  00  t-  t- 
O5  1C  IN  C5  «£> 

oo  os  o  o  —i 


SC-  00  100 
•«f  rj<  IO  t- 
O  t-  r)<  r-l  00 


c-  t-  t-      t-   t-   t-   t- 


CJ  CO  00  OS  N  CO 

T-!  (M  t^    10  CJ  CO 

oo  co  w  ko  co  t- 

oo  <y>  o  1-1  01  co 


"*      •<*  -t 


t-    t- 
t-    t~ 

•*  "* 


S3 


O  t-         U3 

COCOTfTtlT}"  T}«lOl6 

<NC^N(M(N         (MIMCN 


OOOCJO  t-OiOOt-  10 

iHt^N»Ot-  CDt^OJlOt-  CO 

10001000  oort»oo>r-i  TJ< 

T-iocnoooo  OiHcocpt-i  o 


I        I        I 


I  I  I  I  I 


r-l      CSI      CO 

1    I    I 


I  I 


§0000 
0000 
t-  00  OS  O 

I   I    I     I   T 


I    I 


82 

gravitation.  Thus,  by  Table  8,  calcium  (40)  vanishes  at  1000 
kilometers  above  the  photosphere,  that  is,  1".4  arc;  zinc  (65)  at 
675  kilometers  =  0".9;  cadmium  (112)  at  385  kilometers  =  0".5; 
mercury  (198)  at  210  kilometers  =  0".3.  Since  the  radius  of  the 
sun  is  31'  59".26  =  1919".26,  it  follows  that  all  the  gases  whose 
atomic  weight  is  greater  than  40  are  so  distributed  as  to  vanish 
physically  and  materially  at  one  two- thousandth  of  the  radius 
of  the  sun  from  the  plane  of  the  photosphere,  as  defined  by  the 
common  pressure  of  six  atmospheres  for  all  the  elements.  That 
is  to  say,  these  heavy  gases  pass  from  enormous  values  of  tem- 
perature (Table  6),  and  from  enormous  values  of  pressure 
(Tables  9,  10),  to  vanishing  values  in  a  very  narrow  vertical 
stratum.  The  density  changes  (Table  12)  are  by  no  means 
so  surprising,  except  that  all  the  densities  for  the  heavy  gases  are 
really  very  small.  The  sharp  disk  of  the  sun  is,  therefore,  a 
thermodynamic  and  gravitational  effect  upon  gases  of  which  the 
monatomic  weights  are  somewhat  greater  than  the  40  for  cal- 
cium. This  element  just  floats  comfortably  upon  the  layers  of 
still  heavier  gases,  and  it  is  easily  observed  in  the  spectro- 
heliograph  as  a  surface  phenomenon,  having  many  configurations, 
as  in  the  faculae  and  flocculi.  The  lighter  gases,  *  especially 
helium  and  hydrogen,  rise  to  great  heights,  the  hydrogen  to  the 
top  of  the  inner  corona.  The  gases  between  helium  and  calcium, 
up  to  bromine,  columbium,  germanium,  krypton,  molybdenum, 
rubidium,  selenium,  strontium,  yttrium,  zirconium,  ought  to 
just  reach  the  level  of  the  reversing  layer,  if  they  exist  in  the 
solar  envelope.  These  distributions  conform  well  with  the  re- 
sults of  the  spectroheliograph  observations  upon  the  height  of 
the  chemical  elements  in  the  sun. 


CHAPTER  III 

The  Determination  of  the  "Solar  Constant"  of  Radiation  in  the 
Isothermal  Layer  of  the  Sun 

Statement  of  the  Problem 

IN  Bulletins  No.  3,  1912;  No.  4,  1914,  of  the  Oficina  Meteoro- 
logica  Argentina,  and  in  the  Treatise  on  Circulation  and  Radiation, 
1915,  an  account  was  given  of  an  effort  to  solve  the  dis- 
crepancies in  the  data  of  radiation  as  observed  by  the  pyrheli- 
ometer  and  the  bolometer.  The  criticisms  regarding  the  former 
consisted  in  showing  that  in  using  the  Bouguer  formula  for  pure 
radiation  in  the  forms 

(96)  I  =  IQe~km,  or  7  =  To/*, 

for  absorption  and  transmission  respectively,  it  is  not  proper 
to  interpret  the  original  formula, 

(97)  7  =  70£8ec2, 

as  if  sec  z  =  —  were  sec  z  =  m,  for  m^  =  1,  the  depth  of  the 

Wo 

atmosphere.  The  graph  (log  I .  sec  z)  of  observation  serves 
only  up  to  sec  z  =  1  in  the  zenith,  but  it  cannot  be  extended  to 
the  non-mathematical  sec  z  =  o  without  unwarranted  assump- 
tions. These  are,  that  p  is  constant  from  the  sea  level  to  the 
vanishing  plane  of  the  earth's  atmosphere,  whereas  in  fact  p  is 
very  variable;  no  account  is  taken  of  the  potential,  specific 
heat,  and  ionization  energies  which  are  absorbed  to  balance  the 
kinetic  energy,  so  that  the  extrapolated  value  of  70  =  1.950 
gr.  cal./cm.2  min.  depends  upon  a  very  incomplete  treatment 
of  the  observations  made  at  stations  having  different  elevations 
above  the  sea  level;  the  thermodynamics  of  the  radiation 
throughout  the  earth's  atmosphere  shows  that  the  total  amount 
of  black  body  radiation  falling  on  the  outermost  strata  is  about 

83 


84  A  TREATISE   ON  THE   SUN'S  RADIATION 

3.980  gr.  cal./cm.2  min.,  and  that  this  suffers  a  series  of  deple- 
tions which  reduce  it  to  about  1.500  gr.  cal./cm.2  min.  at  the 
sea  level. 

The  data  of  the  bolometer  spectrum  indicate  that  the  sur- 
viving long-  wave  ordinates,  X  =  1.50^  to  X  =  2.50  /x,  seem  to 
originate  in  a  radiating  solar  temperature  of  7700°  Centigrade 
absolute,  while  the  middle  waves,  X  =  0.45  M  to  X  =  1.50  M, 
appear  to  represent  a  radiation  at  about  6900°,  having  lost  a 
large  amount  from  the  original  source,  and  the  short  waves, 
X  =  0.00  M  to  X  =  0.045  /z,  have  evidently  been  excluded  en- 
tirely by  reflection  or  absorption  within  the  high  levels,  40,000 
to  70,000  meters,  of  the  earth's  atmosphere.  The  middle  or- 
dinates, X  =  0.45  n  to  X  =  1.50  fjL,  suffer  progressive  depletion 
from  one  high  station  to  another,  and  arrive  at  sea  level  with 
a  value  of  about  2.47  gr.  cal./cm.2  min.,  when  fully  corrected 
for  band  and  minor  losses  in  the  thermal  spectrum.  These  two 
types  of  observations  afford  such  divergent  data  that  it  is  im- 
possible to  reconcile  them  on  the  hypothesis  that  the  solar 
radiation  originates  in  gases  whose  temperature  is  about  6900°, 
but  whose  efficiency  corresponds  only  to  a  temperature  of  about 
5800°.  This  is,  of  course,  equivalent  to  assuming  that  the 
Kurlbaum  coefficient  of  radiation  a  in  the  Stefan  Law,  at  the 
distance  of  the  earth  D, 


m  •    /-. 

does  not  hold  true  at  the  sun.     Otherwise, 

(98)  7.  =  ,»  (6900)4(|)2  =  a  (5800)4(|)2, 

so  that  th^re  results, 


If  we  take  as  a  mean  value  of  the  Kurlbaum  coefficient 
log  o-  =  —  5.74103,  then  we  should  have  for  the  defective  log  0-1  = 
-  5.44000,  so  that  while  a  =  5.509  X  10~5,  the  defective  or 
inefficient  value  would  be  ai  =  2.754  X  10~5,  half  as  much  as  a. 


SOLAR  CONSTANT"  OF  RADIATION  85 

We  have  already  proved  that  the  mean  temperature  of  the 
sun's  isothermal  layer  is  about  7687°,  as  in  Table  6.  It  only 
remains  to  determine  the  temperature  within  the  isothermal 
layer  at  which  the  emitted  radiation  is  generated,  as  well  as 
the  value  of  <j  actually  prevailing  in  those  solar  strata. 

The  Distribution  of  the  Free  Heat  (Q\  —  Qo) 

The  free  heat  in  any  stratum  (21  —  z0)  is  computed  by  (34), 

(61  ~  Co)  =  (Cpa  -  Cp10)  (Ta  -  To), 

where  (Ta  —  T0)  is  the  adiabatic  fall  of  temperature  in  the 
stratum,  Cpa  the  adiabatic  specific  heat  at  constant  pressure 

for  the   gas,  Cpa  =  Ra  ^£-y,     and     Cp*>  =  J  (Cpl  +  Cpo)  = 

J  C^i  +  Ro) r,  the  mean  of  the  non-adiabatic  values  at  the 

K  —  1 

top  and  the  bottom  of  the  stratum.     In  the  computations  lead- 
ing to  the  adiabatic  value  Ra  there  is  one  fact  of  interest.    At 
the  initial  point,  determined  from  the  terrestrial  values,  we  have 
for  calcium  vapor  on  the  sun,  as  an  example, 
7     X  P  (earth)  =  P  (sun)  =  28.028  X  101323.5  = 

2839900  (M.  K.  S.) 

T  x  T  "  =  T  "  =  28.028  X  273  =  7651.6  (M.  K.  S.) 
y~l  x  p  "  =  p  "  =  1.7999/28.028  = 

0.064219  (M.  K.  S.) 
7  X  R   "   =  R  "   =  28.028  X  206.205  = 

5779.5  (M.  K.  S.) 

This  value  of  R  is  not  the  adiabatic  Ra,  but  only  one  point 
in  the  gas  at  some  distance  above  the  adiabatic  strata.  In 
computing  the  terms  of  the  gravity  equation, 

go  X  28.028  («,  -  so)  =  -  Pl~P°  -  (Cpa  -  CAo)  (r.  -  r0), 

Pio 

this  value  of  (Cpa  -  CplQ)  (7\  -  T0)  =  (Qi  -  Qo)1  is  not  the 
same  as  (Qi  —  Qb)  in  (34),  because  it  is  not  referred  to  the  true 


adiabatic  Cpa  =  Ra     _    .    The  initial  value  of  R  serves  as  a 
preliminary  in  computing  Ra.    Thus,  (Qi  —  Qo)1  is  a  positive 


86  A  TREATISE  ON  THE  SUN'S  RADIATION 

term,  that  is,  it  has  positive  values  which  must  be  added  to  the 

p  p 

positive  value  of -  to  produce  G  (zi  —  z0) ;   below  this 

PlO 

initial  R  we  find  that  (Qi  —  Qo)1  is  a  negative  quantity  to  be 

p  p 

subtracted  from -,  which  has  now  become  greater  than 

PlO 

G  (zi  —  ZQ)  .  Proceeding  in  this  way,  R  enters  at  last  into  a  stratum 
where  it  becomes  a  constant,  Ra,  which  has  a  true  adiabatic  value. 
Now  the  course  of  the  development  of  the  J^-term  depends  upon 
the  trial  temperatures  T  from  level  to  level,  and  if  these  are  not 

exactly  correct  there  will  be  a  corresponding  error  in  'Cpa  = 

% 
Ra 7,  and  all  the  subsequent  terms  derived  from  it.      This 

K  ~~~  L  , 

explanation  is  needed  to  account  for  the  minor  irregularities  that 
occur  in  the  tables  for  eight  elements. 

In  Table  14  the  free  heat  has  been  reduced  to  20-kilometer 
strata,  whatever  may  have  been  the  intervals  in  the  computation, 
for  the  sake  of  comparison  at  different  heights.  Only  a  small 
portion  of  the  data  in  hand  is  here  quoted,  but  this  is  full  in  the 
isothermal  layer  and  near  the  level  of  the  photosphere,  where  it 
is  likely  to  be  of  most  practical  usefulness.  A  similar  arrange- 
ment is  made  for  the  other  tables  of  this  Chapter. 

We  note  that  along  the  lower  boundary  of  the  table  the  value 
°f  (Qi  ~~  (?o)  is  a  vanishing  quantity  in  the  adiabatic  region, 
and  that  along  the  upper  diagonal  a  common  maximum  value  of 
about  6500000  is  reached  for  all  the  elements  whatever  the 
height  may  be  between  the  adiabatic  level  and  the  vanishing 
plane.  The  irregularities  in  this  diagonal  line  depend  upon  the 
facts  that  the  vanishing  plane  does  not  occur  on  the  exact 
height  quoted  in  the  Table,  and  that  the  adiabatic  value  Cpa 
may  not  have  been  found  with  precision  by  the  method  of  trials 
in  this  computation. 

In  Table  14  the  mean  value  of  (Qi  —  Q0)  on  the  photosphere 
is  —  3443372  for  20  kilometers,  or  172.1686  for  one  meter,  this 
being  the  mean  kinetic  energy  or  velocity  square  per  degree. 
It  appears  that  all  the  chemical  elements  on  the  sun  have  the  same 
kinetic  energy  at  the  surface  of  the  photosphere.  Above  that  level 


87 


>oooo     o 

!»iis  i 


t-o 

•>*co 
«ow 
II 


iooi-Hco 


0>       NOS( 


I     II  I     I     I     I     I  I  I     I     I     I     I 


;§  §  g£§! 


o  = 

•^  < 

lilit^ScS      §      CJ^tSJnS      «< 
SScOCN^H       O       OoSrfcoS       00< 

^^^^^     ^     „„„„»     rn< 


0000' 

T}«T}<Ot>l 

WO  t-  -^  ( 


[  I 


TTTTT  7  ^TTT  7  i  i  1 7    i' 


So  o  o< 
COO(M« 


I      I     I      I      I 


co  eo  co  co  eo 

I  I  I  II 


£g§£3     5     SS^^o     6-oco^.o     ^^o^o 
cScSclcS^S      cS      cSclcScSS      co'SSco'S      S^^S§: 

I  I  I  I  I     I     I  I  I  I  I     I  I  I  I  I     I  I  I  I 


j>J     . 


I  I  I  I  I 


§§5§§ 


COCONNCSJ 

COCOCOCOCO 

I  I  I  I  I 


OOOOO  O  OOOOO  OOOOO  0000 

OOOOO  O  OOOOO  OOCOCOO  O-^Ol^OOO 

COOOOO«OT)<  C<J  t-NOOCOO  t-COO5«D^  Wr-HD«)ODOl 

OOO5O5O>  OS  OOOOt-t-«D  Tj<OU5i-lt-  t-  O  «O  US  00  i-H  < 

NNiHiHiH  r-l  rH  i-H  iH  »H  iH  l-H  TH  O  O  OS  t-  CO  t-  »H  N  CO  < 

COCOCOCOCO  CO  COCOCOCOCO  COCOCOCON  N  N  »-l  rH  00  T-I  < 

I  I  I  I  I  I     I  I  II  I  I  I  I  I  I      I  I  I  I  I  I 


I  I  1 


I     II  I 


88 


A   TREATISE   ON   THE   SUN'S   RADIATION 


TABLE  15 
DISTRIBUTION  OF  THE  ENTROPY  (Si—  So)  IN  20-KiLOMETER  STRATA  AT  DIFFERENT  HEIGHTS 

00 

t- 
1 

£s 

CO       CilO^lOt-       t-       OOlO 
OO'       CO  CM  TJ<  CJ  CO*       CO       CMOOTjI 

8^  sssks  w  8°*    ||    |  |  i  i  |    ||    i 

1     IIIII     1     III 

1   1 

S3 

*~:                                     OilO       "*COt-OO>       Cfl       t-OOOO«O 

^                        g  moo     cMcSco.-iS     o     cocot^c^S       III        |        Ml 

s              T7    i  i  i  i  i    i    i  i  i  i  i 

1    1 

** 

IO  CO  IO       -H«  ^<  IO  t~  i—  (       CD       OO  IO  t-  CM  i-l       CO 

ft  osoi-H     ot-«ji,-io>     co     .-it-coot-           1    1    1    1        II        1 
o  tooo     toioioiOTi<     -*     ^cocoeow           1    1    1    1        II        1 

T77    11111    i    11111    i 

1 

QO 

CDCO^t-lO       OOiOt-^       «       t-OOOCM^.       t- 

1  1 

8O  t-  IO  i-H  i-H         r-(O5t-lOCO        CM        OSt-incOiH        00    i      i      |      I             ||             i 
£83;^    ^^^^^    ^   ^.^^^^    ca  |  |  |  |     ||     | 

7  1  i  i  i    11111    i    11111    i 

02 

Oi  •*  T!<  10  00       OCM1000CM       CD       Tji  •<*  TJI  U5  t>       OS  t><3>  IO  CO 

1  1  1 

81OOIOIOIO       «OOrJ<OOeO       t-       OCO5DOiC<J       OSOt^OJO         1     I     1     I 
t~OSCMt-CM       OSOSOOt-t-       CD       CDiO'l'COCO       Ol  N  CM  i-l  O 

1  7  1   1   1       IIIII       1       Mill       IIIII 

i- 

OO       00«OCOOS^       •^'C-OCM'**       t-       t-OOOi-ICM       rjit-IOOCO»o'      CO 

1 

OCMO5       OOJOiOiiH       CMOOOt-iO       CO       O  t>  IO  CM  OJ       IO  Oi  Tf  C<J  iH       CO 
O  00  IO       OOiOcOi-HO       OiO5OOOOOO       00       OOt~t>t-CD       IOCMOOOCD       O    1           1 

THCO     rji^-^^-^      coeoeococo      co      coeocococo     cocoeoiN<N      rn  |        | 

77    11111    11111    i    11111    11111    i 

«o   " 

CM 

CMcoiOi-i©     iocot~»-iio     oooeoiooo      T-I      T-iCMeOTj<io      OSOC<ICDCM      -«teoooio 
>-i^-!dodc<i     t-'oii-J^'cD     CMNi-lda)     oi     odt>'coio^j     ^J^^^^     ^Soi0* 

1 

rj<  CD  N  i-t 

T  i  i  i  i    11111    11111    i    11111    11111    ii    i 

.1 

O  CO  *O  IO  (M         COCOCQC^rH         i-HrHi-Hr-ti-H         iH         T-HrHi-Hr-Ht-^         OOO^C^OO         ^OOCO^ 

t-  T^ 

IOOO 
00 

1 

77  i  i  i    i  i  i  i  i    11111    i    11111    11111    i  i  i  i 

z 

Kilometers 

§0000     ooooo     ooooo     o     ooooo     ooooo     oooo 
OOOO       OOOOO       ©OOCDTl<frl                  <NT}<«DOOO       OOOOO       OOOO 

777 

89 

there  is  an  increase  in  the  value  of  the  kinetic  energy  of  the  free 
heat  per  degree  on  the  same  level,  counting  from  hydrogen  to 
mercury,  while  below  the  photosphere  there  is  decrease  from 
hydrogen  to  mercury  along  the  same  level.  The  former  in- 
creases up  to  the  common  maximum  6500000,  and  the  latter 
decreases  to  the  common  vanishing  value  0.  The  Table  16 
shows  that  (Wi  —  Wo)  =  0  on  the  vanishing  planes,  while 
Table  17  indicates  that  the  inner  energy  (Ui  —  Uo)  reaches  the 
same  maximum  as  (Qi  —  Q0).  Hence, 

(Ci  -  Co)  =  (Sl  -  So)  Tio  =  (Wi  -  Wo)  +  (Ui  -  Uo) 

(Si  -  So)  Tio  =    (Ui  -  Uo)  =  6500000  per  20  km. 

Sl  "  So     &-&)  °° 


Ui-  Uo        6500000   ""  Tio       6500000    "   0' 
since  (Si  —  So)  =  oo  and  7\0  =  0  on  the  vanishing  plane. 

Planck's  formulas,  —  j?  =  —  =  -y,  (Thermodynamik,  135), 

upon  which  the  functions  for  T  the  temperature,  and  K  the 
polarized  specific  intensity,  are  founded, 


that  is  (136),  (137),  (106),  (107),  and  others,  are  really  adiabatic, 
indeterminate  forms  on  the  vanishing  planes  of  gases,  and  are 
not  applicable  within  the  gaseous  atmospheres,  omitting  the 
(Wi  -  Wo)  term. 

It  should  be  noted  that  the  mean  value  on  the  photosphere, 
—  3443372,  is  about  the  mean  between  the  common  minimum 
0  in  the  adiabatic  levels,  and  the  common  maximum  —  6656690 
on  the  vanishing  planes. 

Table  15  for  the  computed  values  of  the  entropy  (Si  —  So) 
in  20-kilometer  strata  shows  that  the  distribution  is  similar  to 
that  for  (Qi  —  Q0),  namely,  a  common  value  —  447.  8  for  all  the 
elements  on  the  photosphere,  an  increase  along  the  several  levels 
from  hydrogen  to  mercury  above  the  photosphere  to  the  maxi- 
mun  oo  ,  and  a  decrease  on  the  levels  below  the  photosphere  to 
the  value  0.  Hence,  the  mean  temperature  of  the  photosphere 


90  A  TREATISE   ON  THE   SUN'S  RADIATION 

from  these  data  becomes  T  =  -  344S372/- 447.8  =  7689.5°, 
while  the  date  of  Table  6  made  the  temperature  7693.1°.  The 
entropy  is  always  negative,  and  it  increases  in  the  vertical 
columns  from  0  on  the  adiabatic  level  to  <*>  on  the  vanishing 
plane.  These  data  are  useful  in  testing  the  theories  of  radiation 
which  have  been  founded  upon  various  hypotheses  and  as- 
sumptions. 

Table  16  gives  the  distribution  of  the  work  of  expansion 
(Wi  —  WQ)  in  20-kilometer  strata,  and  it  is  of  an  exactly  inverse 
model  to  that  of  (Qi  —  QQ)  and  (Ui  —  U0).  The  mean  value  on 
the  photosphere  is  2009412,  and  on  the  levels  above  the  photo- 
sphere there  is  decrease  from  hydrogen  toward  mercury,  from 
a  maximum  to  vanishing  values,  while  below  the  photosphere 
there  is  increase  toward  a  maximum.  The  mean  value  of  this 
maximum  of  work  is  4122458,  and  it  is  the  common  value  of  all 
the  elements  in  the  adiabatic  strata.  It  diminishes  to  0  on  the 
vanishing  planes  of  the  several  gases.  It  should  be  noted  that 
the  mean  value  on  the  photosphere  is  about  the  mean  of  the 
common  maximum  4122458,  which  is,  also,  the  same  as  that  for 
hydrogen,  and  the  common  minimum  0  on  the  vanishing  planes. 

The  distribution  of  the  inner  energy  is  like  that  of  (Qi  —  QQ). 
It  has  the  mean  value  —  5467133  on  the  photosphere;  on  the  levels 
above  the  photosphere  it  increases  from  a  minimum  value, 
namely,  that  of  monatomic  hydrogen,  H\  =  1.00,  at  —  4122458 
to  a  maximum  value  which  is  —  6656690  for  all  the  elements; 
below  the  photosphere  it  decreases  to  the  same  minimum  of 
hydrogen  —  4122458.  It  should  be  noted  that  the  mean  value 
on  the  photosphere  is  about  the  mean  of  the  maximum  —  6656690 
and  the  minimum  —  4122458.  Mean  values  are  applied  to 
individual  gases. 

The  Equation  of  Conservation  of  Energy  has  three  cases: 

The  First  Law  of  Thermodynamics, 

(d  ~  Co)  =  (Wi  -  Wo)  +  (*7i  -  Uo). 

(1)  On  the  vanishing  planes, 

-  6656690  =  0000000  -  6656690. 

(2)  On  the  photosphere,  (approximate) , 

-  3443372  =  2009412  -  5467133. 


SOLAR  CONSTANT"  or  RADIATION 


91 


«o  oooo< 

tO  in>OO^J<0! 

i    ^  t—  i— *  iH  ^  < 

I  co  i-Hcococo- 


§§11 

3IIS' 

CO  CO  CO  CO 


S       55 


oooo 
oom  t- vo 
oo  t~  1000 


O      Ntt^Wt-      00      O 


°°.0.5g! 

COOr)<CO< 


OOOOO 

O  t-OO5Tji       ?D 

C05OU5OOO       T-H 


s§: 


ooooo  ooooo  o  ooooo  ooo 

OrJUOt-Tf       U3Ot-O-*       CO       OICOW5O       IO  «O  t- 


I      oot-ooojoo      t-i-toorHt- 

I         COCOtDTfOO       tH -^  W  OS  CO 


>I-HCO^<O     ooooooooos     Oi     OOOOI-H     co 

(THrHrH^I         rH  iH  T-|  ^H  rH         ^H         iH  CO  CO  CO  CO         CO 


gs 

OrJ«»H  Tl< 

-tfTjuoto     oit-t-^tio     us:  : 

i-HT}"t-O        <OCO«O^fOO        rf 

ooooi-i     co  <o  o  m  o     10 


50OOOO  OOOOO 

IT500U51OO  OtpOiOTf 

It-       CO?O  -. 

COi-IOO5O5  ^<»O«Ot-00       O>       OrHCOCO'*       O5i-(CO 

5DC-OOOOO5  OOOOO       O       i-l  i-l  i-H  i-l  i-l       r-(COrl< 


§§g§    gg^gS    || 


CJ  (N  (N  N  M 


Tj«00 

cococococo     cococococo     coco 


o     ooooo     ooooo  oooooo< 

OiOOCOO       OOOOO  OOOOCOOO< 

SOOCOCO  CO  O  CO  •**  t-  U3  < 

•1-HOrH  O  Tj<  IO  j-4  OS  05  • 

>  o  co  »H  co  o  co  oo  < 

nO  »-H  Tjl  t- i-H  CO  «O  t 

-ICO  CO  CO  CO  CO  CO  CO  t 


ooooo 


ooooo 

I  I  I  I*? 


92 


A   TREATISE   ON  THE   SUN  S   RADIATION 


o  ooooo 

'   C<1  t—  CD  1C  CO  i-H 

«o  in  kom  inio 

I  I    I    I    I    I 


oooo 
co  eo  OS  •<* 

CO  O  t-  N 


»oo  t-  1- 

Tftf  eoeo 
I    I    I    I 


N     5ioo§S?i     S: 

U3        Tj<  rjt  T}<  Tjt  CO       CO 
I  I     I     I     I     I  I 


OOO 
OCOCO 

IO5OO  IO 
OOi-HCO 
CC-^O 
CD  CO  tO 
I  I  I 


ooici     ooco  os  oo  t 


I    I    I    I    I 


oo 

rl<  T* 

II 


:ss°.s  §s 


3  8£S3S8  8S 

OOJOSOIOO        OO  t-tOCOiO-'J1  O>  *-l 

coioicioio     10  m  in  m  m  m  T}"-^ 

I    II    I    I        I  I    I    II    I  II 


O   OOOOO   OOOOO   00 

oooc<i     oowtoo      co  T}< 

t-        6li-IOOiOCN        O5  i-l  N  TJI  CO        O5  O 


ooos      r-imoiooi      t-cDcoioio 

OJt-        CO(N<MT-IO       OOOOO 

ii    ii  M  i    ii  i  i  i    i    i  i  T  i  i" 


_..  .,^§ss  ss 

t-io     t-ococoos      cooo:   :   z 


IS    §§S§§    §§§g§ 


00  O  CO  OO  OO        iH  t*  CO  00  • 

s^^ss  <°^^ 

CO  CO  CD  CO  CD 


5  CO  ! 


ss 


i  mm  mi 


ooooo 

O  CO  O  CO  CO 

o  t-  o  t-  o 

ooeo  w  t-  us 


KSlOkOlO-* 


ooooo 


00 
O  »ft 
OOO       t-  (M 


OO-^OOOOl  "tfCOiHO 

0500t-lOCO  rHrH^H^-l 

10  ic  to  ic  10  10  m  10 10 

II  I  I  I  II  II  I  I  I  I  I  I 


Sooo     ooooooo 
COCMO        j->  05  O  rj<  O  00  CD 
jHCOCDOi        ^OOt-CD0^05 


SOO       OOOOO       O 
>00       OOOCOrflM 


(8J 

J§L 
777 


"SOLAR  CONSTANT"  OF  RADIATION  93 

(3)  On  the  adiabatic  levels, 

0000000  =  4122458  -  4122458. 

The  Second  Law  of  Thermodynamics,  Tw  (Si  —  So)  =  (Qi  — 
go),  on  the  photosphere  is  7689.5  X  -  447.8  =  -  3443372. 

The  results  of  Tables  14,  16,  17  have  been  plotted  on  Fig.  16 
in  order  to  exhibit  the  general  mutual  relations  of  the  free  heat 
(Qi  —  QQ),  work  of  expansion  (Wi  —  WQ),  and  the  inner  energy 
(Ui  —  Uo)j  of  the  different  elements  in  their  distribution  with 
the  distance  in  a  vertical  direction  from  the  level  of  the  photo- 
sphere. Allowance  is  made  for  imperfections  in  the  results  of 
the  computations  by  the  method  of  trials,  the  essential  facts 
being  that  the  lines  pass  through  the  points  indicated  in  the 
mean  values  under  Table  15  at  the  respective  heights  of  the 
top  of  the  adiabatic  layers,  the  plane  of  the  photosphere,  and 
the  vanishing  planes.  The  first  law  of  thermodynamics  is  con- 
served on  every  level,  and  the  heights  as  ordinates  over  any 
value,  taking  in  the  three  terms,  are  distributed  by  the  law 
of  the  equilateral  hyperbola,  x  y  =  c,  already  given.  The 
corresponding  constants  can  easily  be  computed  by  collecting 
the  values  from  the  tables,  but  they  will  be  omitted  in  this  place. 
The  necessary  conclusion  is  that  the  thermodynamic  system 
(Q.  W.  £/.),  as  well  as  the  dynamic  system  (P.  p.  R.  T.),  are 
distributed  according  to  the  exponential  law  of  decay,  counting 
from  the  adiabatic  region  upward.  These  data  can,  therefore, 
be  utilized  in  constructing  the  general  laws  of  solar  physics  in 
their  relations  to  molecular  and  atomic  conditions.  We  shall 
proceed  to  develop  their  consequences  in  relation  to  the  "solar 
constant  of  radiation." 

It  will  be  convenient  to  compute  the  radiation  potential, 

KIQ  =  — —     -  =  c  Ta,   exponent   (a),  and   the  (log  c),  in  the 

Vi         VQ 

law  of  radiation,  Kw  =  c  Ta,  in  order  to  study  their  connection 
with  the  Stefan  Law  J0  =  a  T4. 

The  specific  volume  v  =  l/p  is  computed  from  the  values 
collected  in  Table  12,  and  thence  the  values, 

_  tr.  -  u. 
Kv>-  ~cT  ' 


94 


A   TREATISE   ON   THE   SUN  S   RADIATION 


"SOLAR  CONSTANT"  OF  RADIATION 


95 


I       •       CDOCOCDCD       O        t-  00  O  rH  Ol 
I       I      I      I    iH         rH         CMCO-'tt-rH 


IIII 


I       Mil 


t-     ooooo 

t-       COlO  00<N  CO 

1-1     NOCMCOOO 
I  S     cooo 

I         COOSfN  •«#  t- 

i  1 77^ 


t-     **     as  co  i 


I      III 


OS  O  - 
•IO 

oco 


ooooo     o     ooooo     o 

CD  O  O  O  O       lO        O  Tf  CM  O  CM        •<* 

CMascoajt-     o     t-asrHOco     rH 

O  -^  ^t<  O  lO         rH 

§ 


77777 


t-  rf  IO» 

1OOOCO  O  rH  COCO 

CM  CM  CXI  O  CO  "3CM 

CO    I  I   IN 

I  '  '  I 


I      I  I 


JCDOCMO       OOOOO       O       OOOOO       O       OOO       O 

•  loeoi-HO     ocoi-H-<tcsi     o     wooioot-      ooooo      o 

>OOCOt-Tl<        t-COCOOCO        00       COt-COt-Ol        i-IOOTj<COCO        O 


<  w 


* 


tt~         t- CO  10  •<*  iH         i-H 
S'*        THOS  t-CD  CD       lO 


MM 


OSCO       OO 


CO  OO  t-Ol 

oooooooo  o 

CO-^t1  IOCO  00 


§0000  o  oo 

O  COOCM  t-  COCM 

coas<Nt-csi  t-  ojt^ 

w^t-asoa  ^  loio 

cot-ooas^H  oj  co  ^ 

ooooooooas  as  asasas 


I   M   I   I       I   I   I   I 


t-oot-o     ooooo     ooooo 
oooascoo     Noocot-o     O^^HIOIO 

CDWCOPO        t-rJ-CDNrH        - 


(Ma>cD 

T—  1  CO  CD 


t^  CM  >O  00  T}<  CO  O 

rHcot>cM     t>t-t^6do6     dbdboooooo     60     6606600000     asasasoo     CM »o as  <M rj<  co »o 

I   I   I  l-H    r-(  i-l  t-H  l-l  rH    l-l  rH  rH  r-(  r-(    rH    rH  rH  rH  rH  rH    rH  rH  rH  N  CM    CM  CM  N  CO  CO  •*  lO 


I       I       I 


I       Mill       Mill       I 


M    1    1    1    1    1 


z 

meter 


§o     oo  o< 
o     oooco- 


ooooo 

(N^COOOO 

11117 


I      I      I      | 


1  1  1 


96 

of  Table  18  have  been  found.  KIQ  follows  somewhat  closely 
the  values  of  P  in  Table  9,  only  noting  that  while  P  is  the  value 
of  the  pressure  at  the  height  given,  KIQ  is  the  mean  value  in  the 
stratum  whose  lower  surface  is  the  height  of  P.  Since,  orig- 
inally, 
(tfi  -  Uo)  =  (Ql  -  Qo)  -  (W,  -  Wo)  =  (Qt  -  Q.)  -  P10  (m  -  «,), 

n*  K     Ui~u°   a-&    P 

and  KIQ  — = jPio, 

Vl  —  VQ  VI  —  VQ 

it  follows  that  the  mean  values  for  the  stratum  Kw  and  PIQ 

are  separated  by  the  term  — *.     It  only  remains  to  note 

Vi  -  VQ 

that  the  value  is  always  KIQ  =  0  on  the  vanishing  plane,  ^Tio  = 
—  1990190  on  the  photosphere,  and  that  Kw  increases  to  indef- 
initely large  values  in  the  adiabatic  region  along  with  PIQ. 

Method   of  Computing   the   Coefficients   and   Exponents   in   the 
Formula  of  Radiation 

Taking  helium,  m  =  4.00,  for  an  example  of  the  computed 
values  of  (A,  log  C),  Table  19,  we  note  that  these  immediately 
divide  themselves  into  three  parts,  corresponding  with  the  non- 
adiabatic,  the  isothermal,  and  the  adiabatic  regions;  in  the 
former,  A  approximates  4.00  and  log  C  —  10.00000  more  or 
less;  in  the  isothermal,  A  is  very  large  and  negative,  while 
log  C  is  very  large  and  positive,  due  to  the  large  value  of  n  = 
-  (Ta  —  To)/  -  (Tl  -  TQ)  in  these  layers,  where  the  tempera- 
ture changes  slowly  and  (7\  —  TQ)  is  small;  in  the  adiabatic 
region  A  is  about  2.40  and  log  C  is  about  —  3.20000.  In  order 
to  reduce  these  values  of  (A,  log  C)  to  (#,  log  c)  the  graphs 
(A,  log  C)  are  made  on  Fig.  17,  and  it  is  seen  that  they  separate 
into  two  very  different  distributions.  In  the  adiabatic  layers 
(A,  log  C)  concentrates  about  a  narrow  area,  contained  in  the 
small  square,  while  in  the  non-adiabatic  layers  (A,  log  C)  is 
extended  along  a  straight  line,  from  (A  =  2.00,  log  C  =  0.00000) 
at  the  slope  -  0.35000  for  A  log  C  and  -  0.100  for  A  A. 
The  isothermal  data  can  not  be  plotted  on  a  small  diagram. 
I  have  taken  the  slope  (A}  log  C)  in  the  adiabatic  region  as  if 


OF   RADIATION 


97 


TABLE  19 

EXAMPLE  OF  THE  METHOD  OF  CHANGING  (A,  LOG  C)  INTO  (a  LOG  c)  AND 
LOG  60  GA,  HELIUM 


z 

A 

LogC 

a 

Logc 

Log  60  *A 

(M.  K.  S.) 

(M.  K.  S.) 

(C.  G.  S.) 

11500.  . 

— 

— 

— 

— 

— 

11000.. 

5.6131 

-16.11147 

3.6310 

-11.20850 

-16.36494 

10500.. 

5.7773 

-16.12239 

3.7750 

-11.71250 

-16.86894 

10000.  . 

5.6911 

-16.53935 

3.8474 

-11.96590 

-15.12234 

9500.. 

5.5861 

-16.94719 

3.8927 

-10.12445 

-15.28089 

a 

9000.  . 

5.4619 

-15.38863 

3.9240 

-10.23400 

-15.39044 

O 

8500.. 

5.5085 

-15.28425 

3.9482 

-10.31870 

-15.47514 

u 

8000.. 

5.5506 

-15.17602 

3.9682 

-10.38870 

-15.54514 

.a 

7500.. 

5.4179 

-15.63871 

3.9845 

-10.44575 

-15.60219 

1 

7000.. 

5.1278 

-14.63633 

3.9970 

-10.48950 

-15.64594 

.2 

6500.  . 

4.8676 

-13.54427 

4.0065 

-10.52275 

-15.67919 

"3 

6000.. 

4.6697 

-12.24446 

4.0125 

-10.54375 

-15.70019 

§ 

5500.. 

4.2709 

-11.66218 

4.0178 

-10.56230 

-15.71874 

5000.. 

4.0625 

-10.42998 

4.0205 

-10.57175 

-15.72819 

4500.. 

3.6789 

-8.81261 

4.0198 

-10.56930 

-15.72574 

4000.. 

3.4939 

-8.50161 

4.0185 

-10.56475 

-15.72119 

In 

oj 

3500.. 

3.4420 

-8.70003 

4.0160 

-10.55600 

-15.71244 

X 

3000.. 

3.3830 

-8.92801 

4.0134 

-10.54690 

-15.70334 

£ 

2500.. 

3.4597 

-8.64222 

4.0113 

-10.53955 

-15.69599 

1 

2000.. 

3.0021 

-6.40393 

4.0090 

-10.53150 

-15.68794 

P 

1-1 

1500.. 

4.9972 

-13.10266 

4.0100 

-10.53500 

-15.69144 

M 

1000.  . 

10  415 

-35.68904 

4.0144 

-10.55040 

-15.70684 

i 

500.. 

-130.879 

613.47648 

4.0220 

-10.57700 

-15.73344 

^. 

0.. 

-102.267 

402.82988 

4.0324 

-10.61340 

-15.76984 

-500.. 

-123.089 

484.03055 

4.0402 

-10.64070 

-15.79714 

-1000.  . 

-92.394 

363.42851 

4.0502 

-10.67570 

-15.83414 

-1500.. 

-101.807 

400.96355 

4.0574 

-10.70090 

-15.85734 

-2000.  . 

-  41.316 

166.03588 

4.0658 

-10.73030 

-15.88674 

Sudden    tra 

nsformation    o 

f  the  va 

ues  on  enteri 

ng  the 

adiabatic  st 

rata      % 

-2500.. 

63.4100 

-240.23076 

2.4177 

-3.16071 

-8.31715 

-3000.. 

2.6809 

-4.17847 

2.4252 

-3.17796 

-8.33440 

c 

-3500.. 

2.3819 

-3.35719 

2.4265 

-3.18094 

-8.33738 

o 
'&> 

-4000.  . 

2.3857 

-3.34790 

2.4270 

-3.18210 

-8.33854 

E 

-4500.  . 

2.4080 

-3.26224 

2.4276 

-3.18348 

-8.33992 

.a 

-5000.. 

2.3938 

-3.32427 

2.4280 

-3.18440 

-8.34084 

jj 

% 

-5500.  . 

2.4178 

-3.22897 

2.4285 

-3.18555 

-8.34199 

,Q 

.2 

-6000.  . 

2.4151 

-3.24294 

2.4288 

-3.18624 

-8.34268 

< 

-6500.. 

2.4252 

-3.20316 

2.4292 

-3.18716 

-8.34360 

*s 

-7000.. 

2.4291 

-3.18916 

2.4295 

-3.18885 

-8.34529 

1 

! 

I  . 

! 

1 

I 

Can 

be  extended 

downward    in 

definitely 

98 


A   TREATISE   ON  THE   SUN'S  RADIATION 


it  extended  from  the  same  point  (LOO,  0.00000),  though  there 
may  be  a  question  whether  it  should  not  be  parallel  to  the  non- 
adiabatic  line,  the  shift  being  parallel  rather  than  triangular. 
The  practical  effect  is  the  same  because  (a,  log  c)  extends  for 
so  short  a  distance  in  the  square  that  no  important  change 
could  come  from  a  slightly  different  slope.  The  practical  range 
of  (a,  log  c)  in  the  non-adiabatic  region  is  in  the  small  rectangle. 
Table  20A  contains  the  mean  values  adopted  in  computing 


A 
6.00 

4.00 
200 
0.00 


LogC 


-10.00000 


-20.00000 


FIG.  17.     Helium.     The  (A\  log  C)  Lines  in  the  Adiabatic  and  the  Non- 
adiabatic  Regions. 

the  pair  values  (a,  log  c)  for  helium.  Each  of  the  elements 
HI,  H2,  He,  C,  Ca,  Zn,  Cd,  Eg,  has  been  computed  in  the 
same  way,  and  these  auxiliary  tables  differ  from  one  another 
only  by  minor  variations,  dependent  upon  the  uncertainties 
due  to  the  method  of  trials.  Thus,  collecting  together  the  ini- 
tial values  for  a  =  4.00,  we  have  the  following  results  for  log  c. 
See  Table  19. 


log  A  = 


Formula  (40). 


log  (log  7\  -  log  To)' 
log  C  =  log  KIQ  -  A  log  TV  Formula  (39). 
Plot  (A,  log  C)  and  construct  the  auxiliary  table  for  (a,  log  c). 


OF   RADIATION 


99 


Take  trial  values  of  a  (alt  a2  .  .  .  )  and  compute  log  c  =  log 
KIQ  —  a  log  TIQ]  then  take  the  corresponding  pair  values  (a,  log 
c)  from  the  table.  If  «i  (trial)  =  a  (computed)  the  pair  (a,  log 
c)  is  correct.  If  a\  <  a,  take  #2  =  \  (<*i  +  «)  and  proceed  to  a 
second  trial.  When  au  =  a,  this  is  the  adopted  (a,  log  c). 

These  mean  values  of  Table  20B  might  be  used  to  construct 
a  common  table  for  all  the  chemical  elements,  to  serve  in  a  re- 
computation  based  upon  the  series  of  mean  values  of  the  differ- 
ent hyperbolic  constants  that  have  already  been  obtained.  At 
present,  the  results  given  have  all  been  computed  for  each  ele- 
ment, quite  independent  of  one  another. 

Using  the  auxiliary  Table  20A,  the    pair  values  (a,  log  c) 

TABLE  20A 
THE  WORKING  VALUES  («,  LOG  c) 


Adiabatic  (a  Log  c) 

Non-adiabatic  (ex  Log  c) 

a 

log  c 

a 

log  c 

4.00 

-6.80000 

4.08 

-10.78000 

— 

— 

4.07 

-10.74500 

3.50 

-4.50000 

. 

4.06 

-10.71000 

g 

— 

— 

"So 

4.05 

-10.67500 

as 

3 

2.50 

-3.35000 

c 
a 

4.04 

-10.64000 

§r 

2.49 

-3.32700 

g 

4.03 

-10.60500 

53 

2.48 

-3.30400 

!_ 

4.02 

-10.57000 

aj 

2.47 

-3.28100 

1 

4.01 

-10.53500 

a 

0) 

2.46 

-3.25800 

cn 

4.00 

-10.50000 

J3 

2.45 

-3.23500 

M 

3.99 

-10.46500 

O 

2.44 

-3.21200 

-M 

3.98 

-10.43000 

4-> 

H 

2.43 

-3.18900 

0 

3.97 

-10.39500 

1 

2.42 

-3.16600 

*T3 
0) 

3.96 

-10.36000 

1 

2.41 

-3.14300 

3.95 

-10.32500 

j>» 

2.40 

-3.12000 

3.94 

-10.29000 

2.39 

-3.09700 

J^ 

3.93 

-10.25500 

Jl 

2.38 

-3.07400 

15 

3.92 

-10.22000 

§ 

2.37 

-3.05100 

.y 

3.91 

-10.18500 

1 

2.36 

-3.02800 

2 

3.90 

-10.15000 

</) 

2.35 

-3.00500 

a, 



__ 

4-> 

2.34 

-4.98200 

en 

3.80 

-11.80000 

2.33 

-4.95900 

& 

_ 

_ 

2.32 

-4.93700 

3.70 

-11.45000 

2.31 

-4.91300 

_ 

_ 

2.30 

-4.89000 

3.60 

-11.10000 

Slope 

-0.100 

-0.23000 

Slope 

-0.100 

-0.35000 

100 


A   TREATISE   ON   THE    SUN  S   RADIATION 


TABLE  21 
BUTION  OF  THE  EXPONENT  (a)  IN  Ki0  =  cTa 
ormation  between  the  adiabatic  and  the  non-adiabatic  strata 


DISTR 


The 


s TJI      10  CDOO  -foo      10  t-  ooo  eoiot- 

>  O        rH    NN    NN        CO    N    N  CO    NNN 
IN       N   NN    NN       N   N   NN   NNN 


Non- 
diabatic 
Means 


3 


as 


Z 

om 


•CO        O5    O5    -t  ( 


t~  ( 

to  t-o»  o  t-     oo  oo  oo< 

0>    t-00    0505        05    05    05O 


05NU3010 


i 
TH  CO 

OO 


32  §! 


S§  I  I 


CO   COCO   COCO       CO   CO   CO-* 


I  °. 

CO 


cot>o»oa» 

OOCDCDO5CO 
COCOCOCO-^' 
O5OSO5OSOJ 


oocft 

OOO 
OO  O5 
OS  O5 


coccococo     co     coco 


N     oo5O5Or-i     N     eo-t  10 

O       0050500       O       00   O 


CO-*     •«*  rj<  Tjl  Tjl  •<*  T*  T*  -*•* 


»OiO  ^f  t~  00 

OOO  NCO  IO 

3  COCO  CO 

NN  NN  N 


3  -#    O»   lOr-l  t-  lOOJCOt-O 

5  rf    t-    r-liO  OO  OOi-Hr-IN 

q  r-t  r-i  NN  N  eoeoeococo 

CO-*  -^  -*  rjlTf  TJ 


i-l  CON  t-< 

CD  O5  CO  «D  < 

CO  CO  -4"  ^  » 

o  oo  o < 


°.8§ 

N    NCO 


TH    N 


O5  O5    O}  O5  O5  O5  O5    O5  O5  O5  O5  O5    O5    O5  O5  C5  O5  O5    O5  O5  O5  O5  O5    O>  O 

coco  co'co'  co  co'  coco"  co  co' co  co  coco"  co  co' co  co  coco"  co  co' co  coco"  co"  -*' 


O  N< 
r-l  COI 
10  101 


0?05    0> 


3    CO  CO  CO 
COCO   CO  CO  CO 


O    00t>    U3        Oi-!< 

t-  t-oo  05      eiO( 
oo  oo  oo  oo 


5  O5O5O5O5        O5 


CO   COCO    COCO       CO   CO   COCO   CO       COCOCOCOCO       CO        COCO    CO    COCO       CO    COCO   COCO       •<*    •*    Tjf*    NNN 


II       I       I""*       CO-i'tOOOO       OOOO 
I        I        I      I  I        I      I        I    r-l         N    -*    CDOO 

I       1411,       i     i     i    i 


"SOLAR  CONSTANT"  OF  RADIATIQN 


t>  COO)         CO  OCO  COTjt  00  O  **CO  001O"3 

OS   OSCO         OS   COO  "*00  "5j<    i-H  CO^  COCOCO 

1C    -*t-         OS    OO  OO  t-    CO  COrH  -<}IOOCO 

to   CDt-         t-    OOOS  i-HCO  CO    00  OOO1  t-t-OO 

ICO  COCO  CO    rH  rHrH  rHrHTH 


1 1  sis 

COCOCO         CO    COCO   COCO       CO   CO    COCO    CO  CO  CO 


1C    i-HIO    O    CO 
•^    COCO    1C    O 

-  oo  T}<  co 


10  co  ic 

O  CO     rH  T}" 

t-  t-  coco 

o'  d  o'o  o  d 

7  7  77  7  7 


)  CO  ( 

o'  o'  o  o  o' 

7777  i 


TH    THrH  CO    -*CO    CDt=-  r-i 

t-   t-t-  t-   t-t-   t-t-  CO 

o'o  o  do  d  o'd  do  d  d  o'o' 

77777  77777  7777 


1 1 1 


t-  t- 
co.co 

CO    COCO 

I  I 


u^ 


lCCOOSCDiO 


ic  ic  ic  co 


0 
CO 

ooi-i      ic 
icco     co 

COCO       CO 


SCO    iH    Oil 
OS    CO    CD< 

os  co  oo  co  i 


00   O   O         OOOOO       O 
H  i-H    i-H    TH  THrHi-HTHrH         i-t 

I    I     I     I        Mill       I 


00    O    00 
rHiH    rH    i-HiH 

I   I    I    M 


o  oo 

rH    iH  i-H 

I   I  I 


CO1O        C^ 

Tt<  O  t> 

50^!      £ 


CO  CO        CO 

co'co     co* 

I  I     I 


O  10  O 

OS  t-  ••* 

1C  i-H  O 

co  t-  10 

os  ic  10 


O       1COS   -*f    OOCO 
r}<        CDOO    i-H    CO  CD 
JOJOCO        CO        T}<  1C    t-    OOOS 


CO  00 

So" 


1    1     1 


00    O    O         OOOOO        O       00    O    OO         O    00   OO       O 

HTH  TH  TH    ^^^^^  7  77  7  77   7  77  77  7 


CO  COCO 

I  I  I 


CO  1C  CO 


OSOSOS0300  CO  t-OS   O   i-HCO  O    COCO    00-* 

CDCOOt-^f  i-H  OOlCCOOt-  COr!<CDt-OS 

eoTfiioioco  t-  t-oo  os  oo  TJ<  THOO  icco 

THi-HTHrHi-H  rH  rHrHi-HCOCO  COCOCO^IO 

oo  oo  oo  oc  oo  oo  oooo  oo  oooo  oo  oooo  oooo 


o     o  oo  o  o 

I     I    I  I    I    I 


OOOOO 

7777  i 


oo  o  oo 

77  7  77 


O   00   00       O   O 

77777  77 


-o 


OOt-    CO    1C         ICICIOICIO 
OS  CO  ICOOi-H 

t-       t- 1- 


30  Tj<    ICCO    t-OO  CO    i* 

M  i-H    OOlC   COOS  1C    CO 

^  S££°,8  §^ 

t-t-  t-    t-t-    t-t-  00    00 


O   00   O   O 

77777 


OOOOO       O        00    O    OO 

77777  7  77  7  77 


O   00   00       O   O   00 

i  77  77  7777 


O       O   00   O    O         OOOOO 

I   SSS0.0!    S00^" 


§°.8S§ 
I  I   I  I 


I       I    M    I 


102'  *   TREATISE   ON  THE   SUN'S   RADIATION 


TABLE  20B 

THE  INITIAL  VALUES  OF  a  AND  LOG  c  FOR  THE  DIFFERENT 
CHEMICAL  ELEMENTS 


Element 

Exponent 
a. 

Adiabatic 
Log  c 

Non-adiabatic 
Log  c 

Hydrogen  H\ 

4  00 

—6  80000 

—  10  80000 

Hydrogen  Hz  .... 

4  00 

—6  80000 

-10  95000 

Helium  He  

4.00 

—6  80000 

—  10  50000 

Carbon  C 

4  00 

—7  0500 

—  10  50000 

Calcium  Co,  . 

4  00 

—6  90000 

-10  50000 

Zinc  Zn      .          

4  00 

—6  90000 

-10  50000 

Cadmium  Cd  

4.00 

—7.00000 

-10  60000 

IVlercury  HP 

4  00 

—  7  00000 

-10  99900 

Means                         

4.00 

—6  90556 

—  10  66900 

have  been  computed  by  trials  with  the  results  in  the  fourth 
and  fifth  columns  of  Table  19.  It  is  seen  that  the  very  large 
values  of  (^4,  log  C)  in  the  isothermal  region  have  given  place 
to  the  smooth  values  of  (a,  log  c)  throughout  the  non-adiabatic 
region,  including  the  isothermal  layer.  In  this  region  the  value 
of  the  exponent  increases  slowly  from  3.6310  at  z  =  11500 
kilometers  to  4.0658  at  —  2000  kilometers,  while  log  c  changes 
from  —  11.20850  to  —  10.73030  at  these  respective  heights, 
counted  from  the  level  of  the  photosphere.  In  passing  into 
the  adiabatic  levels  a  sudden  and  important  transformation  occurs, 
such  that  the  exponent  a  changes  from  4.0658  into  2.4177,  and 
the  coefficient  log  c  changes  from  -  10.73030  into  -  3.16071. 
That  is  to  say,  in  the  adiabatic  region  we  have  (a,  log  c)  = 
(2.4177,  —  3.16071),  and  in  the  non-adiabatic  layers  at  the 
bottom  of  the  isothermal  layers  we  have  (a,  log  c)  =  (4.0658, 
—  10.73030).  The  radiation  terms  therefore  pass  through  a 
saltum  at  the  bottom  of  the  isothermal  strata. 

Table  21  contains  the  values  of  the  exponent  a  on  the  se- 
lected levels,  and  it  shows  the  same  remarkable  transition  from 
values  a  little  greater  than  4.00  to  about  2.4200.  There  is  one 
conspicuous  exception  to  this  rule,  namely,  hydrogen  as  a 
diatomic  molecule,  H2  =  2.00,  where  the  transformation  of  the 
exponent  is  from  4.0094  to  3.3323  instead  of  to  2.4200.  This 
pronounced  difference  of  the  behavior  of  the  diatomic  element 


OF   RADIATION  103 

H2  from  the  monatomic  HI  is  very  significant.  It  has  been 
necessary  to  assume  that  HI  =  1.00  is  the  condition  which  is  to 
be  used  in  the  adiabatic  levels,  while  the  coronal  observations  dem- 
onstrate that  hydrogen  reaches  its  normal  limit  at  the  top  of  the 
inner  corona,  and  must  therefore  have  become  H2  =  2.00  above 
the  level  of  the  transformation.  This  is  interpreted  as  the  evidence 
of  the  dissociation  of  hydrogen  in  the  adiabatic  strata. 

Table  22  shows  that  there  is  the  same  transformation  of 
log  c  in  all  the  elements,  from  about  —  10.78000  to  about 
—  3.20000.  The  exception  again  is  diatomic  hydrogen,  H2  = 
2.00,  where  the  transition  is  from  -  10.98083  to  -  7.77434, 
instead  of  to  —  3.20000,  which  is  again  to  be  associated  with  the 
process  of  dissociation  at  the  levels  —  4000  to  —  6000  kilometers 
below  the  level  of  the  photosphere.  Tables  21,  22,  indicate  plainly 
that  the  level  of  the  transformation  is  always  near  the  bottom 
of  the  isothermal  layer,  at  the  depth  determined  by  the  hyper- 
bolic law,  and  that  it  is  connected  with  the  entire  set  of  physical 
processes  at  these  levels,  which  are  contained  in  the  preceding 
tables.  The  mean  values  of  (a,  log  c)  in  both  regions  are  given 
in  the  last  columns  of  the  Tables  21,  22. 

The  u  Solar-Constant"  of  Radiation 

On  Table  22  the  transition  between  the  values  of  log  c  at 
the  bottom  of  the  isothermal  layer  appears  to  stand  on  a  diag- 
onal straight  line,  but  this  is  merely  due  to  the  selection  of  the 
vertical  height  that  has  been  employed.  As  a  matter  of  fact, 
this  boundary  forms  an  equilateral  hyperbola  according  to  the 
law  z  m  =  c.  By  interpolating  the  probable  depths  below  the 
plane  of  the  photosphere  where  this  transition  takes  place,  it 
is  found  that 

ZR  .  m  =  -  9000. 

By  Table  7,  the  bottom  of  the  isothermal  layer  is  ZA}  so  that 

ZA  .  m  =  —  12000. 
Hence,  for  each  different  element  we  obtain  the  depth  of  ZR 


104 


A   TREATISE   ON  THE   SUN  S   RADIATION 


for  radiation,  and  of  ZA  the  depth  of  the  transition  to  adiabatic 
conditions. 

TABLE  23 
POSITION  OF  THE  RADIATION  LAYER  ZR  IN  THE  ISOTHERMAL  LAYER 


Element  m 

ZA 

ZR 

Zl 

zT=Q 

Hi    1.00  

-12000 

-9000 

+3600 

+45400 

H2    2  00 

—  6000 

—4500 

+  1800 

+22700 

He   4 

-3000 

—2250 

+900 

+11350 

C   12   

—  1000 

-750 

+300 

+3780 

Ca   40 

—300 

—225 

+90 

+1135 

Zw   65 

—  185 

—  140 

+55 

+700 

Cd  112 

—  107 

—80 

+32 

+405 

Hg  198 

—60 

—45 

+  18 

+230 

ZA  =  the  bottom  of  the  isothermal  layer.     Table  5. 
ZR  =  the  depth  of  the  radiation  layer.     Tables  21,  22. 
Zi  =  the  top  of  the  isothermal  layer.     Table  7. 
ZT=O  =  the  height  of  the  vanishing  plane.    Table  8. 

The  position  of  the  radiation  stratum  in  the  midst  of  the 
isothermal  layer  appears  to  be  at  three-fourths  of  the  distance 
of  the  bottom  of  the  isothermal  layer  ZA  from  the  photosphere. 
Compare  figures  2-9.  On  Fig.  18  the  hyperbolic  curves  for  the 
constants  at  the  top  of  Table  23  show  the  relative  positions 
of  ZA,  ZR,  ZT,  ZT=O)  corresponding  with  the  atomic  /weights 
of  the  chemical  elements.  The  asymptotic  condition  is  appar- 
ently reached  for  the  heaviest  chemical  elements,  Radium  226, 
Thorium  232,  Uranium  238. 

Mean  Values  of  the  Coefficient  and  the  Exponent  of  Radiation 
in  the  Stefan  Law 

The  computations  show  that  at  whatever  depth  the  radia- 
tion transformation  may  occur  for  the  different  elements,  the 
preceding  and  the  following  values  have  only  small  changes  as 
compared  with  the  abrupt  transition  itself.  They  may,  there- 
fore, be  collected  together  without  regard  to  the  values  of  z  for 
which  the  computations  were  actually  made,  that  is,  for  the 
arbitrarily  selected  intervals  (zi  —  ZQ).  These  were  chosen  in 
order  to  make  fifty  or  sixty  points  in  each  computation,  and 


OF  RADIATION 


105 


they  varied  approximately  in  the  following  intervals:  H2  2000, 
Hl  500,  He  200,  C  200,  Ca  50,  Zn  25,  Cd  20,  Eg  10  kilometers, 
respectively,  for  the  (zi  —  z0)  strata. 


75 


100 


125 


150 


175 


FIG.  18.     Four  Typical  Hyperbolic  Curves. 
The  hyperbolic  curve  for  m  ZT=O   —  +45400,  Vanishing  Plane. 
The  hyperbolic  curve  for  m  zx        =  +  3600,  Top  of  Isothermal  Layer. 
The  hyperbolic  curve  for  m  ZR       =  —  9000,  Radiation  Stratum. 
The  hyperbolic  curve  for  m  ZA       —  — 12000,  Bottom  of  Isothermal  Layer. 

Table  24  contains  the  values  of  (a,  log  c)  for  the  last  ten 
values  in  the  isothermal  layer,  1,  2,  ...  10,  and.  the  first  ten 
values  in  the  adiabatic  strata,  10,  9,  ...  1.  The  transforma- 
tion of  these  values  takes  place  in  the  radiation  layer  which  lies 


106 


A   TREATISE   ON  THE   SUN  S   RADIATION 


0 

00  CO  »O  CM  IO  O  CO 

S  £H  O  ^H  O  i-l  CO 

osoooooS 

CO  •<*  •"*'  •<*'  ^'  •*  CO 

t- 

1 

CO 

00  W  1C  O  IO  CO  O 
OJ  OJ  OJ  CT>  0>  0)  0> 

1  1  1  1  1  1  1 

os 

00 

os 

1 

- 

OS 

rH  O  IO  •*  OS  CM  IO 
CO*  **  Tj<  TJ<  rji  Tj<  CO 

CO 

W  0  10  0  10  i-l  CO 

«o  co  co  co  •«*  eg  05 
co  10  10  »o  10  ib  t- 

O  O5  O5  O5  OS  Ci  Oi 

1  1  1  1  1  1  1 

co 

co 

s 

co 

OS 

1 

CM 

00 

c<i  o  TJI  oj  as  t-  oo 

<£>OOCOOOO(M 
OJOOOOOiS 

i 

o 

Ol  O  O  »O  IO  i-l  CO 

oo  10  co  co  co  10  05 

t^  CO  CO  »O  t»  OS  O 
«O  tO  lO  lO  »O  U3  00 

IO 
CO 

CO 

<! 

CO  T*  ^  Tj  -#  CO  CO 

CO 

OS  OJ  0  <7»  OS  O>  OJ 

11  1  1  1  1  1 

OS 

1 

1 

U 
« 

en 

I 

t- 

a>  Tf  oo  co  o  N  o> 

OJ  ^  ^  00  O5  00  *& 
CO  •*  •*'  TjJ  Tj<'  CO  CO 

CO 

/—  s 

"n? 

Tj<OOOOiOO 

^-*oo  10  10  coo 

CM  O  i-H  •*  i-l  CO  CO 
O  lO  lO  CO  O  O5  CM 
t-  lO  lO  lO  CO  lO  00 

CS  GS  C7S  C75  OS  OS  OS 

1  i  1  1  1  1  1 

»0 

s 

CSJ 

CO 

OS 

1 

•** 

S 

1 

thermal  la 

Q0 

13 

^H  O  tfi  50  t-  t>  <0 

N  N  CO  'S*  lO  •**  CO 
OO  N  O]  W  CO  O  iO 
OSOOOOOOi 

CO  Tjt  i*  Tjt  -<t  Tt  CO 

•^ 

(isotherm. 

«OOlO  O  lO  O  TJI 

U5  O  (M  T-H  OS  CO  CO 

CO  t^  00  K>  S  i-<  f 

os'  os'  oi  o>  o>  os'  os' 

1  1  1  1  1  1  1 

^J< 

IO 

s 

CO 
OS 

1 

2 
X 

4o 

*4-» 

s 

IO 

4 

SI  S  o 

8 

!s 

10 

« 

4-> 

j3 

lllllsi 

o 
o 

o 

I 

rf  O  O  lO  VO  pa  t- 

T*  Tf  lO  Tj<  C-  lO  CO 

U3  CO  OS  OO  lO  O  00 
CO  iH  CM  CS1  lO  lO  t- 

t-  CO  CO  CO  CO  CO  00 

i 

oj 

1 

_c 

CO 

N    H  OQ 

3  -d 

a 
£ 

0) 

1 

CO  -^  -^  "*  •*  "t  CO 

•^ 

1 

OS  OS  OS  OS  OS  OS  O> 

1  1  1  I  1  1  1 

OS 

1 

4-> 

d 

«  ~ 

JSB^ 

2 

'£ 

VJ 

£ 

T3 

T* 

£ 

CD 
,C 

*o 

ss^^sss 

SSSScigg 

TjJ  Tjl  Tf  Tjl  Tj<  -^  CO* 

S 
S 

•Tf 

CG 

1 
•S 

NOOOO^Ht- 

lO  C-  CM  O  CO  OO  CO 
tM  O  kO  Tf  i-l  O  OS 

o  •**  co  >o  oo  os  os 

00  CO  CO  CO  CO  CO  00 

os'  os'  os  os'  os  os'  os 

1  1  1  1  1  1  1 

t- 

os 
c- 

os 

1 

<o 

° 

-o 
q 

rt 

t- 

i 

cd 
8 

•8 

CO 

1 

•«* 
to 

s 

en 

(1) 
3 

1 

5 

o> 

s 

.^ 

en 
V 

3 

00 

% 

<j 

| 

P—  1 

•*' 

M 

1  1  1  1  1  I  1 

1 

1 

3 

0 

£ 

t-  Tj<  O  t-  t-  ^O 

c-  1^  as  a>  -«f  ifl  oo 

T-H  Irt  CO  1C  CO  ^1"  00 

CO 

•^" 

O3  O  O  to  lO  kO  O 

6-  os  io  os  -^  co  o 

CO  O  ^f  O  C<J  ^O  CO 

CM 

I 

ooooooos 

o 

, 

en 
H 
P 

Tjl  Tj<  •*}!  Tt  Tjl  TX  CO 

T}< 

OS  OS  OS  OS  OS  OS  OS 

i  1  1  1  1  1  1 

OS 

1 

1 

Sllggil 

(O 

i 

CO  O  O  O  10  t-  Tji  . 
t~  CO  IO  t-  CO  CO  CO 

co  o  co  o  CM  oo  co 
oo  co  oo  co  io  oo  os 

00  t-  t-  t-  t-  t-  OS 

1 

00 

0 

Tj«  Tf  Tj<  Tj<  Tjl  -^  CO 

•* 

OS  OS  OS  OS  OS  OS  OS 

1  1  1  1  1  1  1 

OS 

1 

b 

b 

£ 

jp 

s 

3 

s 

3 

£ 

2 

»-!  Til  e<i  o  w  esi  oo 

THTfCO^HOi 
i-t  rH 

^^OUNU^ 

Means.  . 

rH  •«*  N  O  lO  N  00 

^HTfCO^OS 
rH  »H 

^^UONUte! 

Means.  . 

" SOLAR  CONSTANT"  OF  RADIATION 


107 


co  t-  10  eo  co  co  10 

eo  o  10  oo  co  CD  co 

o 

CO  t-  rf  O  CO  O  IO 

Tf  0  CD  CO  00  00  rH 

t-  CD  CO  CO  T}<  IO  IO 

rH  rH  CO  CO  CO  CO  CO 

os 

•^  co 
^  co 
os  t> 

il 

CO  CO  CO  CO  CO  CO  CO 

3 

co  co  co  co'  co'  co  co 

1  1  1  1  1  1  1 

CO 

1 

eo'os 
1 

1 

•*  co  o  eo  10  co  t- 

rH  IO  CO  00  t~  IO  OS 

CO  CO  CO  CO  rH  O  OS 

CO 

t- 

co  co  10  os  m  T}<  t- 

COOS  O  O  CO  O  OS 
St-  Tf  U5  O  O5  U5 
t-  TJ<  oo  co  t-  co 

rH  rH  CO  CO  CO  CO  CO 

IO 

<f 

IO 
CO 

OS 

Tj< 

CO  t> 
CO  t* 

CO  CO  CO  CO  CO  CO  CO 

co  co  co  eo  co  co  co 

1  1  1  1  1  1  1 

eo 

1 

coos 
I 

coco 

1     . 

10  10  o  t—  eo  co  rH 

1 

rH  rH  CO  CO  CO  CO  CO 

5 
i 

CD 
IO  t- 

s§ 

os  oo 

co 

rH  CO 

eooo 

«««««"" 

« 

CO  CO  CO  CO  CO  CO  CO 

1  1  1  1  1  1  1 

CO 

1 

coos' 

1 

coco 

1 

8S3S8SB 

«! 

t-  0  CO  »0  OS  0  rH 

OO  rH  O  t-  CO  IO  00 

rH  rH  CO  CO  CO  CO  CO 

to 

i 

CO 

CO 

lOrH 

IO  rH 

OS  00 

SCO 
03 

*iM«N««N 

^__^ 

CO  CO  CO  CO  CO  CO  CO 

1  1  1  1  1  1  1 

CO 

1 

<N 

coos' 

1 

coco 

1 

IO  CO  OO  t-  rH  CO 

3 

1 

.2 

lOOOkOOrHOrH 
CO  -<f  >O  CO  CO  OS  00 

CO  CO  t-  rH  CO  t-  rH 

oo  oo  co  os  t-  oo  t- 

rH  rH  CO  CO  CO  CO  CO 

co 

S 

o 

IOIO 

£§8 

II 

CO  CO  CO  CO  CO  CO  CO 

CO 

•6 

v^S 

to 

CO  CO  CO  CO  CO  CO  CO 

i  1  1  1  1  1  1 

eo 

1 

a 
s 

y 

co  os 

1 

coco 
1 

IO  O  CO  OS  CO  OO  00 

CO  CO  CO  CO  CO  O  O 

fl 

jf 

w 

IO  O  t-  t-  CD  rf  Tf 

co  TJI  os  eo  eo  us  10 
oo  oo  §  os  c^  oo  c^ 

1 

a 

c 

f 

4U 

g 

w 
>< 

ON 

H 

coco 

KNNNNNN 

M 

O) 

•g 

bE 

CO  CO  CO  CO  CO  CO  CO 

1  1  1  1  1  1  1 

CO 

1 

.0 
O 

_1 

H) 

3 

coos' 

1 

y 

coco 
1 

t-  IO  CO  00  •>*  T*  O 

t-  oo  10  co  co  o  co 

CO  CO  CO  CO  CO  -H  O 

o 

CO 

I 

g 

rH  10  co  •*  co  eo  t- 

eo  10  co  *o  co  o  co 

1 

2 

^ 

k 

5THERM 

II 

i 

s 

§1 

CO  CO  CO  CO  CO  CO  CO 

CO 

4-> 

•o 

CO  CO  CO  CO  CO  CO  CO 

1  1  1  1  1  1  1 

eo 

1 

03 
oJ 

a 

coos' 

1 

•5 

coco 
1 

O  00  IO  IO  OO  CO  t~ 

oo  oo  co  co  eo  rH  -«f 

CO  CO  CO  CO  CO  rH  O 

§ 

1 

13 
> 

co 

-o 

0) 

cfl 

53 

t- 

•t  eo 
co  oo 

NNNNNWN 

> 

co  co  co  co  eo  co  eo 

1  1  1  1  1  1  1 

CO 

1 

c^ 

coos 

1 

coco 
1 

•*  CO  000000 
00  OS  iO  -^  rfi  O  TJ< 

co  co  rf  eo  co  rH  o 

3 

CO  rH  CD  CO  CO  O  O 
IO  t-  t>  OO  IO  O  CD 

rH  rH  CO  CO  CO  CO  CO 

1 

M 

> 

t- 

eo 

eo  oo 

•w«p««52 

" 

co  co  eo  co  eo  eo  co 

1  1  1  1  1  1  1 

CO 

1 

co'os 

1 

coco 
I 

lOlOOt-rHOO 
CD  O>  O  Tj<   Tf  O  Tj< 

IO»OO  rH  COO  O 

os  ooo  oo  Tji  o  o 

O  00  TJ<  OJ  IO  O  CO 

oo  oo  oo  01  1-  os  c- 

rH  rH  CO  CO  CO  CO  CO 

1 

t- 
co 

O  00 

oos 

cow 

eo'eo'NNeo'eo'co' 

co  co  co  co  eo  co  co 

1  1  1  1  1  1  1 

eo 

1 

1 

coco 

1 



::;:;;; 

'.'.'.'.'.'.'. 

I-H  ^t  co  o  m  co  oo 
»U  &3  U  U  N  O  &3 

1 

s 

rH  TP  CO  0  10  CO  00 

rHT)«COrHOS 

s 

«;! 

vi 
M 

108  A  TREATISE   ON  THE   SUN^S   RADIATION 

among  them  at  various  depths.  Section  I  contains  the  mean 
values  of  the  exponent  a  in  the  isothermal  layer,  and  it  ranges 
from  4.0526  to  3.9937,  approximately  4.00  on  the  average; 
Section  II  contains  the  coefficient  log  c,  which  ranges  from 
-  9.81409  down  to  -  9.60739  (C.  G.  S.);  Section  III  contains 
the  values  of  the  exponent  a  in  the  adiabatic  layer,  and  it  ranges 
from  2.4241  down  to  2.4110;  Section  IV  contains  the  values  of 
the  coefficient  log  c,  and  it  ranges  from  —  2.27072  down  to 
—  2.24019  (C.  G.  S.) ;  Section  V  contains  the  corresponding  values 
for  the  diatomic  hydrogens  H2  =  2.00,  and  it  shows  that  the  system 
(a,  log  c)  is  the  same  in  the  isothermal  layer,  but  that  it  is  very 
different  in  the  adiabatic  strata,  since  the  exponent  a  ranges  from 
3.3522  down  to  3.3213  and  the  coefficient  log  c  from  —  6.84068 
down  to  -  6.73767  (C.  G.  S.).  The  minus  sign  affects  only  the 
characteristic  of  the  logarithm. 

An  examination  of  Table  24  indicates  that,  although  there 
are  minor  differences  among  the  elements,  there  prevails  a  gen- 
eral harmony  in  all  respects.  The  same  range  of  the  mean 
values  is  found  for  each  element.  Such  differences  as  exist 
depend  upon  the  inaccuracies  with  which  the  adiabatic  Ra  and 
Cpa  were  computed  by  the  method  of  trials.  The  most  im- 
portant data  are  the  adiabatic  values  just  before  (2.1)  entering 
the  radiation  layer,  and  those  values  which  occur  just  within 
(1.2)  the  isothermal  layer.  In  the  values  of  log  c  of  the  adia- 
batic strata,  the  hydrogen  HI  and  helium  He  are  apparently  a 
little  too  small  in  comparison  with  the  other  elements.  It  is  be- 
lieved that  an  adjusted  computation,  following  the  hyperbolic 
constants  throughout,  will  produce  a  mean  value  under  (l)  about 
like  that  now  under  (2).  The  mean  —  2.25450  has,  therefore, 
been  adopted  for  the  following  discussions  regarding  the  value 
of  the  solar  constant  of  radiation.  The  general  problem  is  to 
interpret  the  following  data: 

Isothermal  Layer,  a  =  4.00,  log  c  =  -  9.81400  (C.  G.  S.). 

Adiabatic  Strata,   a  =  2.4172,  log  c  =  -  2.25450     " 

The  sudden  transition  from  the  adiabatic  to  the  isothermal 
conditions  seems  physically  like  the  release  of  a  powerful  spring, 
the  discharge  of  a  cannon,  or  a  radioactive  process,  in  which  the 


109 

compressed  state  of  the  adiabatic  strata  is  per  saltum  transformed 
into  the  weak  state  of  the  isothermal  layer,  the  coefficient  c  being 
one  three-millionth  less  in  value.  It  is  assumed  that  this  represents 
the  process  of  transformation  in  the  configuration  of  the  electrons, 
atoms,  molecules,  the  change  of  state  being  accompanied  by  a  power- 
Jul  radiation  into  space.  This  is  the  source  of  the  solar  constant 
of  radiation,  whatever  its  history  of  depletion  may  become.  We 
have  therefore  to  evaluate  the  mean  value  of  the  coefficient 
log  c,  of  which  the  limits  have  been  determined,  supposing  this 
mean  value  to  be  the  a  of  the  Stefan  Law,  /o  =  <r  7\ 

In  making  the  evaluation  of  the  mean  log  c0  it  is  not  proper 
to  take  the  mean  value  of  the  extremes,  J  (log  c\  +  log  cz),  be- 
cause this  assumes  that  the  process  of  decay  or  discharge  is 
arithmetical.  On  the  other  hand,  all  analogies  suggest  that  the 
process  of  depletion  must  be  logarithmic.  We  must  assume  that 
this  process  begins  as  a  tangent  to  a  logarithmic  spiral  on  the 
level  log  o-i  =  —  2.25450,  and  ends  as  a  normal  to  this  curve 
at  log  o-2  =  —  9.81400.  We  have  to  determine  the  angle  of  a 
logarithmic  spiral  whose  origin  is  on  the  axis  of  ordinates,  which 
passes  —  2.25450  as  a  tangent,  and  becomes  perpendicular  to 
—  9.81400.  Such  a  logarithmic  curve,  r  =  ea9,  has  the  angle 
a  =  35°  10',  so  that  a  =  cot  a  =  1.4198.  It  will  be  convenient 
at  this  point  to  collect  together  the  analytic  conditions  of  a 
logarithmic  spiral. 

Logarithmic  Spiral 

(102)  General  Equation.        r  =  ea&.  r  =  radius,  6  =  angle, 

a  =  constant. 

(103)  Differentials.  ^  =  a  ea°  =  a  r.    ^  =  e00  (1  +  a2)* 

a  u  a  u 

=  r  (1  +  a*)". 
dr_  _         a 
ds~  (1  +  a*)*' 

(104)  Element  of  the  curve  ds  =  (d  r2  +  r2  d  0)*  = 

a*)"  dd=  (l  +  *2)*dr. 


110  A  TREATISE   ON   THE   SUN'S   RADIATION 

(105)  Trigonometric  functions. 

_rde  1 

:  ~J7  : 

dr 


__ 
=  Ts  =  (1  +  a*)*' 

rdS       I  dr 

tan  a  =  —  ;  —  =  —  .     cot  a  =  —  j-r  =  a. 
dr        a  r  d  6 


(106)  Tangent  and  Normal.     T  =  r  ^  =  —  (1  +  a2)*. 

d  r       a 

tf  =  ^  =  r(l  +  0*)«. 

(107)  Subtangent,  Subnormal. 

S.T.  =  r^  =  L.       5JV.  =  ^  =  a 
dr       a  dd 

(108)  Perpendiculars  on  T,  N. 

dd  r  dr 

Pt  =  r2d~S==(l  +  a*)*-    Pr  =  TTS 

ar 


-  (1  +  «' 

r  f         d  r\  ** 
(109)  Line  integral.     (si-s0)  =  J  (r*  +  j^    d0  = 


ine  integral.     (SI  —  SQ)  =  (l  +  -^—2)  (ri  —  r0)  = 


Line 

(110)  Area  or  surface  integral. 

A  = 

(111)  Volume  of  revolution. 


(112)  Center  of  gravity,     go  =  Y2 

y  dx 

da 

Compare  the  International  Cloud  Report,  page  515. 


Ill 


The  Logarithmic  Spiral  for  a  =  35°  10' 

The  angle  35°  10'  is  the  constant  angle  at  which  the  logarith- 
mic curve  crosses  the  radius  from  the  pole  to  any  point  on  the 
curve.  See  Fig.  19.  The  computations  proceed  by  the  following 
formulas : 

a  =  35°  10',  cot  a  =  a  =  1.4193. 

Compute  the  value  for  6  =  45°  =  — ,  log  r  =  M  a  —  b. 

Take  the  factor  b  as  a  series  of  multiples  of  45°,  5  =  0,  0.2, 
0.4,  0.6,  0.8,  1.0,  1.1  ...  3.5. 

TABLE  25 
POINTS  ON  THE  LOGARITHMIC  SPIRAL  OF  35°  10' 


r 

e 

r 

e 

r 

e 

Deg. 

Deg. 

Deg. 

0.2000 

0.0 

.0626 

67.5 

3.2594 

112.5 

0.2499 

9.0 

.1901 

72.0 

3.6282 

117.0 

0.3024 

18.0 

.3303 

76.5 

4.0562 

121.5 

0.3904 

27.0 

.4873 

81.0 

4.5346 

126.0 

0.4878 

36.0 

.6628 

85.5 

5.0690 

130.5 

0.6097 

45.0 

.8588 

90.0 

5.6666 

135.0 

0.6816 

49.5 

2.0780 

94.5 

6.3348 

139.5 

0.7620 

54.0 

2.5230 

99.0 

7.0822 

144.0 

0.6518 

58.5 

2.5970 

103.5 

7.9170 

148.5 

0.9523 

63.0 

2.9032 

108.0 

8.8510 

153.0 

The  point  of  tangency  to  —  2.25450  must  occur  at  the  angle 


e  =  90°  -  35°.167  =  54°.833, 
must  occur  at  54°.833  +  90°  = 


and    the 
144°.833. 


normal   to    —  9.81400 


fri  =  0.77790  and  j  r2 
pair  values  are,   |     =  ^^          j 


=  7.23000, 
=  144°.833. 


On  the  scale  of  the  drawing  the  values  of  r\,  r2  are  such  that 
the  unit  is  1.00000  on  the  logarithm  of  the  ordinates.  Take  the 
line  integral  from  $1  (n,  0i)  to  s2  (r2,  #2),  and  we  have,  (s2  —  Si)  = 
7.8926.  Hence,  the  middle  point  is  at  (SQ  -  si)  =  3.9463.  The 
pair  values  for  the  middle  point  are  r0  =  4.0040,  00  =  120°.98. 
This  is  on  the  ordinate  for  log  a0  =  —  5.74000  by  the  drawing. 


112 


A   TREATISE   ON   THE   SUN  S   RADIATION 


Log  (9 


-8, 


-6. 


-5. 


-3. 


-2. 


-1. 


00000 


ooooo 


80 


40 


OOOOO 


80 
OOOOO 


ooooo 


80 
OOOOO 


20 
40 
60 
80 
OOCOO 


10 
60 
80 

OOOOO 


0.778  fi 
54?8  0i 


Log  #0-9.8  400 


Log 


Log  6 


5.74000 


-2.25450 


7.230  rj 
144=8  02 


'4.0045 


The  35  10  logarithmic  spiral  r=el 
a  =  cot  35°10'=  1.4198 


FIG.  19.     The  Mean  Coefficient  of  Radiation  O-Q  in  the  Stefan  Law,  Jo  =  O-Q  T"4 

Resolving  the  component  (n  .  00,  0.7779  cos  54°.  101,  0.45540 

(ro  .  fc),  4.0040  cos  59°.0,  2.06220 

Sum  of  the  two  component  ordinates  (o-i  to  <70),  2.51760 

Value  of  the  log  <n  at  the  initial  point,  —  2.25450 

"       "     "   log  (70  "    "    middle  point,  -  5.73690 

"      "     "   log  o-2  "    "    final  point,  -  9.81400 


OF   RADIATION 


113 


The  result  is  that  the  coefficient  of  radiation  corresponding 
to  the  middle  point  of  the  logarithmic  curve  is  —  5.73690  by  the 
analysis,  and  —  5.74000  by  scaling  from  the  drawing. 

It  only  remains  to  compare  this  result  with  the  values  of  the 
Kurlbaum  coefficient  in  the  Stefan  Law  as  determined  by  the 
laboratory  experiments.  The  following  values  are  collected  in 
the  (M.  K.  S.)  and  (C.  G.  S.)  systems  of  units.  The  dimensions 
of  o-  in  this  place  are  the  same  as  those  of  equation  (38)  and  its 
equivalents,  (M.  K.  S.)  X  10  =  (C.  G.  S.),  for  the  computed  a. 
This  is  to  be  distinguished  from  60  a  /A ,  the  equivalent  in  gram 
calories  per  square  centimeter  per  minute.  (Table  19.)  The 

coefficient  a  =  — ,  as  may  be  seen  in  the  formula  for  spatial 

density  of  the  energy  of  radiation  by  Formula  (76)  and  others 
following.  As  it  is  not  proper  to  press  too  strongly  the  accuracy 
of  the  details  of  such  a  computation,  I  have  provisionally  adopted 
log  a  =  —  5.74000  (C.  G.  S.),  as  in  the  following  collected  results. 

TABLE  26 

VALUES  OF  THE  KURLBAUM  COEFFICIENT  IN"/O  =  <T  T4=— —    -^- .  -^-  =  7—    . 

4    L  T2      L3      LT2J 


Authority 

M.  K.  S. 

C.  G.  S. 

Log  <r 

ac 
"=-* 

Log  <r 

Loga 

Log  60  <rA 

cm.2  min. 

Planck  
Schwarze  
Berlin  Phys.  Inst. 
Westphal  
Shakspear  
Bigelow  

-6.72393 
-6.73480 
-6.73719 
-6.74370 
-6.75358 
-6.74000 

5.2956X10-6 
5.4300     ' 
5.4600     ' 
5.5424     ' 
5.6700     ' 
5.4954     ' 

-5.72393 
-5.73480 
-5.73719 
-5.74370 
-5.75358 
-5.74000 

-15.84887 
-15.85974 
-15.86213 
-15.86864 
-15.87852 
-15.86494 

-11.88037 
-11.89124 
-11.89363 
-11.90014 
-11.91002 
-11.89644 

7.  5922X10-H 

7.7847     " 
7.8276     " 
7.9458     " 
8.1287     " 
7.8784     " 

The  value  of  60  aA  (C.  G.  S.)  adopted  in  Bulletins  No.  3, 
No.  4,  O.  M.  A.,  and  in  the  Meteorological  Treatise,  page  280,  is 
7.90  X  10"11. 

From  this  summary  it  is  at  least  safe  to  infer  that  the  coefficient 
of  radiation  in  J0  =  a  T4  is  the  same  in  the  solar  atmosphere,  in 
the  special  layer  of  radiation  described ,  that  it  is  in  the  terrestrial 
laboratories. 

We  have  already  given  the  temperatures  in  the  isothermal 
layer,  Table  6,  the  mean  value  in  that  collection  being  7686°.7. 


114 


A  TREATISE   ON   THE   SUN'S   RADIATION 


It  is  found  in  the  computations  that  in  the  isothermal  layer  of 
each  gas  there  is  a  gradual  small  decrease  in  the  temperature  of 
each  element  from  the  top  to  the  bottom  of  these  strata,  just  as 
there  is  in  the  earth's  isothermal  layer,  40000  to  12000  meters 
above  the  sea  level.  We  therefore  take  from  the  computations 
the  values  of  the  temperature  of  each  element  at  the  depth  below 
the  plane  of  the  photosphere  assigned  to  the  radiation  stratum, 
where  the  abrupt  transformations  of  Table  24  take  place. 

TABLE  27 
THE  TEMPERATURE  IN  THE  RADIATION  LEVELS 


Element 

Depth 
in 

Kilometers 

Temperature 
of 
Radiation 

Hi          1  00 

-9000 

7656.6 

H2        2  00  

-4500 

7645.9 

He        4     

-2250 

7658.3 

C        12 

—750 

7655.0 

Ca      40         .                                .            

-225 

7658.3 

Zn      65       

-140 

7657.0 

Cd    112 

—80 

7651  0 

Hg    198 

—45 

7653.6 

Mean 

7654.5 

The  mean  temperature  of  radiation  is  7654°.5  absolute. 
If  the  radius  of  the  sun  is  R  =  694800  kilometers,  log  5.84186, 
and  the  distance  to  the  earth  is  D  =  149340900  kilometers,  log 

(Tj  v    _ 
—  J  =  —  5.33536  is  the  reduction  factor  to 

the  mean  distance  of  the  earth  for  the  solar  constant  of  radiation. 

60 
In  the  (C.  G.  S.)  system  the  factor 


In7 

AU 


=  0.0000014336, 


log  —  6.15644.  We  compute  the  values  of  the  radiation  by  the 
Stefan  Law, 


(113) 


at  the  sun  and  at  the  distance  of  the  earth. 


TABLE   28 
EVALUATION  OF  THE  FORMULA, 


to  determine  the  "  solar  constant  "  of  radiation. 


Formula 

Sun 

Earth 

T 

0 

7654  5 

o 

7654  5 

Los  T 

3  88392 

3  88392 

Log  T4.  . 

15  53568 

15  53568 

Log  (T 

—5  74000 

—5  74000 

Log  60/4.  1851  X107.... 

(R\* 
Log  1  —  1 

-6.15644 
0  00000 

-6.15644 
—5  33536 

1  \DJ 

j      gr.  cal. 

5.43212 
270470 

0.76748 

5&544  Solar  Cnn^tflni- 

cm.2  min. 

At  the  sun  the  emission  of  radiation  is  270470  gram  calories 
per  square  centimeter  per  minute,  and  at  the  distance  of  the  earth, 
without  any  depletion,  this  is  equivalent  to  5.8544  gr.  cal./cm.2  min. 

The  usual  value  assigned  to  the  solar  constant  of  radiation 
by  the  observers  of  the  Smithsonian  Institution  is  about  1.940 
gr,  cal./cm.2  min.,  as  determined  by  the  pyrheliometer  when 
the  reductions  are  made  by  the  Langley-Abbot  method  of  ex- 
trapolation for  a  special  assumption  in  the  Bouguer  formula. 
By  the  thermodynamics  of  the  earth's  atmosphere,  Bui.  No.  4, 
O.  M.  A.,  it  was  found  that  3.980  gr.  cal./cm.2  min.  is  the  effec- 
tive amount  received  and  utilized  in  the  atmosphere  in  order  to 
maintain  the  existing  temperatures,  pressures,  densities,  gas  effi- 
ciencies that  actually  exist.  We  must,  therefore,  infer  that  the 
amount  lost  by  scattering  or  internal  reflection  and  by  absorp- 
tion between  the  radiation  layer  and. the  vanishing  plane  in 
the  sun's  atmosphere  amounts  to  about 

5.85  -  3.98  =  1.87  gr.  cal./cm.2  min., 

which  is  about  one-third  of  the  original  amount.    Hence,  we 
may  summarize  briefly  from  Bui.  No.  4: 


116 


A  TREATISE   ON  THE   SUN'S   RADIATION 


Depleted  by  scattering  or  absorption  at  the  sun 1.87 

earth....  2.48 

Received  on  the  sea  level  by  the  pyrheliometer 1.50 

Total  solar  radiation.  .  .  .5.85 


TABLE  29 

THE  AVERAGE  PHYSICAL  CONDITIONS  IN  THE  SOLAR  STRATA  WHERE  THE 
RADIATION  ORIGINATES 


IN  20—  KILOMETERS 

Depth 

T 

A 

Element 

in 
Kilo- 
meters 

Tem- 
pera- 
ture 

Pres- 
sure 

At- 
mos- 
pheres 

p 

Density 

R 
Gas 

Coefficient 

Qi-Qo 
Free 

Wi-Wo 

Work- 

Ui-Uo 
Inner 

Heat 

Energy 

Hi     1.00 

-9000 

7656.6 

2426700 

23.950 

.  0014609 

216950 

-978630 

3164920 

-4156600 

Hz     2.00 

-4500 

7645.9 

2429700 

23  .  980 

.  0028738 

110580 

-1498200 

3912000 

-5433120 

He     4. 

-2250 

7658.3 

2432050 

24.003 

.  0058514 

54273 

-598870 

3475200 

-4095440 

C     12. 

-750 

7655  .  0 

2587250 

25.535 

.018218 

18553 

-1856200 

3418780 

-5428500 

Ca  40. 

-225 

7658.3 

2421600 

23.900 

.  058363 

5418 

-1173310 

3407620 

-4711480 

Zn  65. 

-140 

7657  .  0 

2567800 

25.343 

.  098010 

3422 

-1142800 

3336560 

-4513140 

Cdll2. 

-80 

7651  .  0 

2839900 

28.028 

.  17902 

2073 

-599590 

3525580 

-4125270 

H0198.  .  . 

-45 

7653.6 

2493280 

24  .  047 

.  29056 

1096 

-686080 

3286780 

-4042200 

7654.5 

2524785 

24  .  848 

.  0014889 

220772 

-1066710 

3440930 

-4563218 

Xm=  p 

m 

It  is  desirable  to  know  the  average  dynamic  and  thermody- 
namic  conditions  which  prevail  in  the  solar  strata  where  the 
thermodynamic  radiation  has  its  source.  That  is  to  say,  those 
are  the  physical  conditions  which  are  suitable  to  set  in  operation 
the  readjustment  of  the  electrons  in  the  atoms  and  molecules. 
They  are  as  follows: 

Temperature 7654°.5  A. 

Pressure  in  kilograms  per  square  meter  2524785 

Pressure  in  terrestrial  atmospheres 24.848 

Density  in  kilograms  per  cubic  meter.  0.0014889  X  m 
Gas  coefficient  in  velocity  square  per 

degree 220772/w 

The  free  heat  in  work  per  cubic  meter  —  1066710  (20  —  kilom.) 

The  work  of  external  expansion 3440930 

The  inner  energy -  4563218  " 


"SOLAR  CONSTANT"  OF  RADIATION  117 

Dividing  the  last  three  by  20000,  we  shall  have  the  work  or 
kinetic  energy  per  cubic  meter,  Joules  per  m3. 


(Qi  -  <2o)  the  free  heat  ...........  .  .    •-  53.3355 

(Wi  -  Wo)  the  work  of  expansion.  .  .  .         172.0465 
(Ui  -  UQ)  the  inner  energy  .........    •-  228.1609      " 

Multiply  by  10  to  obtain  ergs  per  cubic  centimeter. 
These  values  must  be  incorporated  into  the  theory  of  the 
physics  of  solar  radioactive  radiation. 


The  Evaluation  of  Ja  =  a  T^—CQ 


With  the  computed  values  of  log  c  and  a,  as  in  Tables  19 
to  22,  the  value  of  the  absorbed  energy  in  the  several  strata 
has  been  obtained.  For  convenience  of  comparison  the  log  c 
has  been  transformed  to  log  60  <*A  C.  G.  S.  in  order  to  have 
the  result  in  gr.  cal./cm.2  min.  As  it  is  not  practical  to  tran- 
scribe the  entire  computation,  the  values  of  Ja  have  been 
summed  in  strata  as  indicated  in  Table  30  under  z  in  kilometers 
for  each  element.  The  sum  S  Ja  for  each  stratum  is  placed 
against  the  Zi  for  the  top  of  the  stratum,  the  bottom  being  the 
next  value  ZQ  below  it.  The  approximate  position  of  the  top 
of  the  isothermal  layer  zly  the  photosphere  000,  the  radiating 
stratum  ZR,  and  the  top  of  the  adiabatic  .layer  ZA  are  indicated 
on  the  Table  30.  The  depths  of  the  strata  differ  from  element 
to  element,  so  that  the  classification  is  complex.  It  is  found, 
however,  by  taking  the  sum  S  Ja  from  the  vanishing  plane 
down  to  the  adiabatic  strata  that  the  amount  of  energy  in  these 
strata  averages  94.46  gr.  cal./cm.2  min.  for  each  chemical  element. 
This  is  the  amount  of  energy  required  within  the  gases  to  main- 
tain the  existing  solar  T.  P.  p.  R.  and  the  prevailing  thermo- 
dynamic  conditions.  In  the  earth's  atmosphere  this  amounts 
to  about  2.47  gr.  col.  /cm.2  min.  down  to  the  sea  level  in  middle 
latitudes. 

From  the  sums  S  Ja,  and  the  heights  Zi  —  Zo,  the  value  of 
A  J  =  S  JJ  (  zi  —  Zo)  gives  the  gradient  of  the  heat  contents 


118 


A   TREATISE   ON   THE   SUN  S   RADIATION 


per  kilometer.  When  plotted,  these  gradients  form  a  fan-shaped 
diagram  from  small  values  on  the  top  line,  close  to  the  abscissas 
of  the  atomic  weights,  up  to  a  high  diagonal  within  the  adia- 
batic  strata  (lowest  line  on  the  table).  These  gradients  are 
essential  in  a  theory  of  the  solar  radiation.  It  should  be  noted 
that  while  (a,  log  c)  differ  abruptly  in  passing  through  the 
radiation  stratum,  Table  24,  it  yet  appears  that  the  gradient  of 
the  heat  contents  does  not  change  per  saltum,  so  that  radiation  does 
not  seem  to  be  a  common  thermodynamic  process,  as  treated  by 
Planck. 

TABLE  30 

EVALUATION  OF  THE  ABSORBED  ENERGY  2  Ja  =  2  (a  7Y*i  —  c0  7Y*o)  AS  A 
TOTAL  IN  SELECTED  STRATA,  AND  AS  A  GRADIENT  OR  AVERAGE  AMOUNT  PER 

KILOMETER 


HYDROGEN  1.00 

HYDROGEN  2.00 

HELIUM  4.00 

CARBON  12.00 

Z 

2  Jo 

A  Ja 

Z 

2Ja 

Aja 

Z 

2Ja 

A  Ja 

Z 

2  Ja 

Aja 

48000... 

0.026 

.0000 

25000 

0.002 

.0000 

11000 

0.143 

.0001 

4000 

0.233 

.0001 

34000... 

0.622 

.0005 

17000 

0.143 

.0018 

7000 

2.813 

.0025 

2400 

4.005 

.0040 

24000.  .  . 

3.723 

.0012 

12000 

1.901 

.0026 

4000 

16.358 

.0043 

1400 

19.717 

.0150 

140'JO... 

14.648 

.0018 

7000 

12.845 

.0034 

1000 

4.029 

.0058 

400 

6.013 

.0200 

Zi 

Zi 

Zi 

Zi 

4000... 

3.982 

.0020 

2000 

4.147 

.0036 

500 

4.349 

.0068 

200 

6.352 

.0240 

2000.  .  . 

3.546 

.0020 

1000 

3.556 

.0038 



.0075 



.0300 

000.  .  . 

5.100 

.0021 

000 

4.321 

.0046 

000 

3.775 

.0077 

000 

*  7.342 

.0380 

-2000... 

4.320 

.0022 

-1000 

7.335 

.0052 

-500 

4.682 

.0085 

-200 

7.207 

.0420 

-4000... 

6.132 

.0023 

-2000 

8.151 

.0064 

-1000 

4.600 

.0100 

-400 

11.211 

.0500 

-6000... 

4.701 

.0032 

-3000 

8.344 

.0080 

-1500 

6.196 

.0126 

-600 

12.064 

.0600 

-8000.  .  . 

8.178 

.0040 

-4000 

11.281 

.0100 

-2000 

9.131 

.0180 

-800 

6.997 

.0800 

ZR 

ZR 

ZR 

ZR 

-10000... 

5.140 

.0052 

-5000 

12.903 

.0129 

-2500 

7.648 

.0240 

-1000 

11.744 

.1000 

.0068 

—6000 

12.740 

.0180 

-3000 

28.214 

.0320 

-1200 

27  .  606 

.1850 

'ZA  ' 

ZA 

ZA 

ZA 

-12000... 

17.458 

.0087 

-7000 

50.280 

.0251 

-3500 

40.075 

.0401 

-1400 

89  .  720 

.4000 

-14000... 

22.436 

.0112 

-9000 

92.020 

.0420 

-4500 

64.480 

.0620 

-1800 

156.070 

.6804 

-16000... 

26.794 

.0133 

-11000 

131.740 

.0658 

-5500 

81.550 

.0816 

-2200 

226.310 

1.1316 

-18000 

—  13000 

-6500 

-2600 

To  ZA     60.118 

87.669 

91.938 

120.491 

SOLAR  CONSTANT       OF   RADIATION 
TABLE  30 — Continued 


119 


CALCIUM  40 

ZINC  65 

CADMIUM  112 

MERCURY  198 

Z 

2  Jo 

A  Ja 

Z 

2  Ja 

A  J0 

Z 

2Ja 

AJa 

Z 

2  Ja 

A  Jfl 

1000  
700  
400  
100  

50  

0.117 
2.993 
16.480 
4.835 
Zi 
5.394 

.0004 
.0250 
.0500 
.0750 

.0800 
.0850 
.0900 
.1100 
.1320 
.1700 
.2200 

.3750 
.5048 

.8250 
1.3020 
1.8670 

675 

475 
275 
75 

50 
25 
0 
-25 
-50 
-75 
-100 

-125 
-150 

-200 
-250 
-300 
-350 

0.078 
2.707 
17.484 
0.792 
Zi 
5.342 
3.025 
6.177 
2.695 
2.255 
6.670 
6.052 
ZR 
6.026 
9.100 

ZA 

38.241 
49.430 
62.480 

.0008 
.0450 
.0710 
.1000 

.1150 
.1300 
.1400 
.1600 
.1850 
.2300 
.3100 

.4300 
.7000 

1  .  0500 
1.5500 
2.1000 

360 
240 
140 
40 

20 

0.156 
2.819 
18.211 
4.297 
Zi 
5.381 

.0013 
.0700 
.1500 
.1950 

.2250 
2500 

200 
140 
80 
20 

10 

0.111 
3.087 
30.505 
4.382 
Zi 
4.979 

.0020 
.2000 
.3500 
.4250 

.4500 
.4700 
.4900 
.5250 
.6000 
.7200 
.9000 

1  .  1500 
1.5109 

2.3867 
3.0670 
3.9055 

0  
-50  
-100 

3.758 
6.274 
4.589 
11.431 
9.773 
ZR 
11.028 
54.478 

ZA 

172.570 
260.340 
373.390 

0 
-20 
-40 
-60 

-80 

-100 
-120 

-140 
-180 
-220 
260 

5.037 
6.668 
6.388 
3.654 
8.523 
ZR 
18.563 
27.313 

ZA 

69.990 
80.830 
104.430 

.2700 
.2920 
.3200 
.3900 
.5150 

.7400 
1  .  0600 

1.4500 
2.0507 
2.7108 

0 
-10 
-20 
-30 
-40 

-50 
-60 

-70 
-90 
-110 
—  130 

2.939 
5.872 
4.524 
8.038 
2.048 
ZR 
7.308 
15.109 

ZA 

47.733 
61.340 
78.110 

-150  
-200  

-250  
-350  

-450  
-650  

-850  
-1050  

To   ZA           131.150 

68.403 

107.010 

88.901 

Zl  =  the  top  of  the  isothermal  layer. 
ZR  =  the  radiation  stratum. 
ZA    =  the  top  of  the  adiabatic  layer. 

2  Ja  =  the  sum  of  the  absorbed  energy  in  the  stratum  down  to  the  next  value  of  Zo  (Zi-Zo) 
A  Ja  is  the  average  amount  in  a  stratum  1000  meters  deep. 

The  sums  at  the  bottom  of  2  Ja  are  the  total  amounts  from  the  vanishing  plane  down  to 

gr.  cal. 


the  top  of  the  adiabatic  layer  ZA.     The  average  amount  is  94.46 
distance  of  the  earth. 


cm.2  min. 


reduced  to  the  mean 


NOTES,  (i)  R.  Emden,  in  his  Uber  Strahlungsgleichgewicht  und  atmospharische  Strahlung," 
1913,  page  63,  arrives  at  this  theorem:  "An  isothermal  atmosphere  which  is  sufficiently  thick 
emits  black  radiation."  The  solar  atmospheres  conform  to  this  criterion. 

(2)  In  the  Appendix  to  this  Treatise  a  more  specific  analysis  of  the  physical  conditions  at 
the  layer  of  radiation  confirms  the  saltum  process  which  is  described  in  this  chapter. 

(3)  It  seems  that  radiation  begins  suddenly  at  the  bottom  of  the  isothermal  layer,  and  is 
continued  throughout  these  strata  as  black  or  fully  effective  radiation. 


CHAPTER  IV 

The  Coefficients  in  the  Stefan  and  the  Wien-Planck 
Formulas  for  Black  Body  Radiation 

The  Conversion  from  the  (M.  K.  S.}  System  to  the  (C.  G.  S.) 

System 

UP  to  this  point  the  computations  have  all  been  made  in 
the  (M.  K.  S.)  system  of  units,  because  for  the  data  of  meteor- 
ology it  is  much  more  convenient  in  many  respects.  The  gram 
and  the  centimeter  are  too  small  for  practical  purposes  in  the 
atmosphere,  and  their  use  would  entail  a  set  of  inconvenient 
and  unmanageable  numbers.  On  the  other  hand,  in  the  magni- 
fied correlative  system,  the  ton  =  1003  grams  =  1000  kilograms 
is  a  unit  of  mass  too  large  for  the  practical  man,  while  the  unit 
of  length,  1  meter  =  100  centimeters,  is  very  suitable.  The 
unit  of  time,  one  second,  is  the  same  in  both  systems.  The 
best  compromise  for  general  purposes  is  the  meter-kilogram- 
second  (M.  K.  S.)  system,  which  has  been  employed  in  the 
computations. 

In  taking  up  the  problems  of  radiation  the  convenient 
system  is  evidently  the  (C.  G.  S.)  system,  because  the  entire 
literature  of  the  physics  of  radiation,  including  the  funda- 
mental constants  and  coefficients,  is  written  on  that  basis. 
Accordingly,  the  remainder  of  our  discussion  will  be  executed 
in  the  (C.  G.  S.)  system  of  units.  In  making  the  transition 
from  (M.  K.  S.)  to  (C.  G.  S.),  the  equation  of  condition  for 
problems  in  radiation  is  that  heretofore  employed,  and  it  is 
derived  from  the 
First  Law  of  Thermodynamics,  dQ  =  dW  +  dU. 

Since  d  W  =  P  d  v,  this  transposes  and  integrates  into, 

(ft  -  ft)        p          Vi  -  0.  _.  K-         fTa 

—, v /io  =  =  A 10  =  C  L  10   . 

(Vi  -  Vo)  V!  -  VQ 

120 


COEFFICIENTS  IN  STEFAN  AND  WIEN-PLANCK  FORMULAS      121 

This  equation  has  the  dimensions  of  kinetic  energy  per  unit 
volume  in  every  term,  and  it  represents  the  work  done  in  devel- 
oping the  kinetic  energy  of  the  moving  molecules  within  the 
given  volume.  Thus,  (Qi  —  Qo)  is  the  free  heat  which  is  due 
to  the  velocity  of  the  molecules  in  the  volume  (vi  —  VQ).  Since 

(ft  -  Q.)  =  (cp.  -  Q>10)  (T.  -  r0), 

L2 
it  must  have  the  dimensions  of  the  specific  heat,  ^—7 — ,  the 

dimension  of  (vi  —  VQ)  =  -rjrJ    therefore,  — °  has  the  di- 

JVL  Vi  —  ^o 

.         D          M  M        fa-m 

mension  ^-3 —  .  -77-  =  r  ^9  , —  (M.  K.  S.). 
T2  deg.      U       L  T2  deg. 

kilog.  meter2     ML2      ...  . 

For   1   Joule  = = -^ — ,      ^2   ,   this  is  equivalent    to 

kilogram  Joule  Joule 

meter  second2  degree       meter3  deg.       volume  deg.' 

ST.  cm.2 
Similarly,  in  the  (C.  G.    S.)    system,  for  1  erg  =  — — p, 

(?i  —  Qo ,  gram          erg       dyne 

-  becomes  -2 9  =  — ~  =  -11— ;. 

PI  —  VQ  cm.  sec.2      cm.3       cm.2 

M 
Since  -j-^-  =  10,  we  have,  (M.  K.  S.)  X  10  =  (C.  G.  S.)  and 

the  conversion  factor  is  10  from  (M.  K.  S.)  to  (C.  G.  S.). 

PIO  is  the  pressure  as  work  per  unit  volume,  and  not  force 
per  unit  area,  though  the  dimensions  are  the  same, 

Pressure  =  work  per  unit  volume  =  M  L2T~  .  L   =M  L~  T~ . 

Pressure  =  force  per  unit  area  =  M  L  T~*.  L~2  =  M  IT1  T*. 

Its  conversion  factor  from  (M.  K.  S.)  to  (C.  G.  S.)  is  10. 
(Ui  —  UQ)  is  the  inner  energy  per  unit  volume,  due  again  to  the 
kinetic  energy  of  the  molecular  motions.  Since  (Ui  —  Uo)  = 
(Cpa  -  Rio)  (Ta  -  T0),  it  foUows  that  the  same  factor  of  con- 
version applies  as  for  (Qi  —  Q0). 

Kio  is  the  energy  of  the  electromagnetic  radiation  per  unit 
volume,  and  it  is  so  denned  by  all  authorities.  Since  TIQ  and  the 
exponent  a  are  without  dimensions,  it  follows  that  the  coefficient 
c  is  the  energy  per  (degree)**  in  the  unit  volume. 


122  A   TREATISE   ON  THE   SUN'S   RADIATION 

It  may  be  noted  in  passing  that  Mr.  Frank  W.  Very,  of  the 
Westwood  Astrophysical  Observatory,  insists  upon  interpreting 
this  equation  of  condition  as  if  it  were, 

Joules  .  ,  Joules   , 

-  --  instead  of  ~,  --  (M.  K.  S.). 
area  volume 

Very>  JouIes=Joule?=kilog1^    J      [Ml,  Factor=1000. 
area          m2  sec.2         m2  '  LpU> 

Joules       Joules      kilog.  m2     1     [~  M  ~| 

Bigelow,  -iL-i  --  =*-  —  -  —  =  —  ^—  .  —  -,     ^—  -    ,  Factor=10. 
'volume         m3  sec.2       m3   LL  PJ 

He  has  published  an  erroneous  criticism  regarding  my  com- 
putations in  asserting  that  the  conversion  factor  is  1000  in- 
stead of  10.  This  is  evidently  contrary  to  the  common 
law  of  pressure  in  meteorology,  which  converts  the  pressure 


P  =  101323.5    2i£^?  into  1013235.  ess*.  and  not  into 

meter2  cm.2  ' 


101323500  -         .     That  would  destroy  the  entire  meteorological 

system  in  practice. 

It  may  be  proper  in  this  connection  to  make  a  few  quotations 
as  to  the  meaning  of  u  —  a  T*  when  referring  to  the  Stefan 
formula;  KIQ  =  cr  T4  evidently  has  identical  dimensions. 

Lorentz,  "The  Theory  of  Electrons,"  page  74.  "<r  is  a  con- 
stant. The  total  energy  per  unit  volume,  or  the  total  emissivity  of 
a  black  body,  must  be  proportional  to  the  fourth  power  of  the 
temperature." 

Arthur  L.  Day  and  C.  E.  Van  Ostrand,  "Astrophysical 
Journal,"  January,  1904,  page  4.  "  d  Q  =  differential  of  radiant 
heat  entering  the  cylinder,  d  W  =  the  external  work,  d  U  =  the 
increase  of  internal  energy.  /0  =  U/u  =  the  density  of  the  energy 
or  intensity  per  unit  volume" 

Richardson,  "The  Electron  Theory  of  Matter,"  p.  334.  "It 
follows  that  the  energy  per  unit  volume,  in  vacuo,  of  radiation  in 
equilibrium  in  an  enclosure  at  the  absolute  temperature  T  is 
equal  to  a  universal  constant  multiplied  by  the  fourth  power  of 
the  absolute  temperature." 


COEFFICIENTS   IN   STEFAN  AND    WIEN-PLANCK  FORMULAS   123 

Planck,  "Die  Theorie  der  Warmestrahlung," 
"P.  62,  u  =  specific  intensity  of  the  black  radiation. 
"  P.  23,  u  =  the  volume  density  of  the  total  radiation. 
"  P.  63,  u,  the  volume  density  and  the  specific  intensity  of 
black  radiation,  are  proportional  to  T*. 

"  P.  63,  «  =  7.061  X  10-'5 


volume 

This  is  the  energy  in  a  volume  in  the  pure  ether  assumed 
to  be  enclosed  in  a  vessel  of  perfectly  reflecting  mirror  walls,  or 
in  which  the  vessel  is  maintained  at  a  certain  temperature  T. 

This  confined  radiant  energy  when  put  in  motion  with  the 
velocity  of  light  c  becomes  the  radiating  energy  in  one  direction, 
so  that  a  column  having  the  base  of  1  cm.2  and  the  length  c  is 
the  amount  whichj  as  a  radiating  flux,  can  enter  an  enclosure  per 
second.  If  this  is  a  sphere  the  area  of  the  surface  is  TT  r2  X  4, 

and  that  of  the  area  on  the  diameter  is  TT  r2,  so  that  a  =  —  a  is  the 
amount  of  the  flux  which  crosses  a  unit  area  per  second  in  one 

direction.     Hence,  J0  =  a  T*  =  —  T*  is  the  surface  flux  of  black 

a 
radiation,      o-  =  —  X  velocity.      This  is   the   simple  interpre- 

tation of  the  Poynting  Law  which  connects  the  surface  flux 
with  the  volume  density  (66). 

Planck  defines  the  Kirchhoff  Theorem  in  the  following  terms, 
page  133: 

"A  vacuum  surrounded  by  mirror  walls  in  which  arbitrary 
emitting  and  absorbing  bodies  are  distributed  in  an  arbitrary 
arrangement,  in  the  course  of  time  come  to  the  stationary  state  of 
black  radiation,  which  is  fully  determined  through  one  parameter, 
the  temperature,  and  therefore  does  not  depend  upon  the  number, 
the  kind,  and  the  arrangement  of  the  ponderable  bodies. 

The  general  problem  of  radiation  consists  of  three  distinct 
parts  : 

(1)  The  electromagnetic  black  radiation  confined  for  con- 
venience in  a  mirror  enclosure. 


124  A   TREATISE    ON   THE    SUN'S   RADIATION 

(2)  The  ponderable  bodies  or  gases  in  the  same  enclosure 
having  any  distribution. 

(3)  The  stationary  state  of  equilibrium  at  the  temperature 
Tj  between  the  electromagnetic  radiant  energy  and  the  kinetic 
energy  of  molecular  motion,  brought  about  by  the  mechanism 
of  the  molecules,  atoms  and  electrons,  which  are  involved  in 
the  conservation  of  energy  for  the  system,  during  the  trans- 
formation from  the  radiant  to  the  kinetic  motions  of  ponderables. 
Without  knowing  what  the  structural  mechanism  really  is,  there 
is  the  transformation  of  a  definite  amount  from  the  radiation 
energy  per  volume  into  the  temperature  corresponding  with  the 
given  kinetic  energy. 

Lorentz  expresses  the  general  problem  as  follows  in  terms 
of  Planck's  resonators,  on  page  79.  "If  a  body  is  enclosed 
within  perfectly  reflecting  walls,  there  will  be  a  state  of  equilib- 
rium, on  the  one  hand  between  the  resonators  and  the  radia- 
tion in  the  ether,  and  on  the  other  hand  between  the  reson- 
ators and  the  ordinary  heat  motion  of  the  molecules  and  atoms 
constituting  the  ponderable  matter.  The  first  of  these  equilib- 
ria can  be  examined  by  means  of  the  electromagnetic  equa- 
tions, and,  in  order  to  understand  the  second,  one  could  try 
to  trace  the  interchange  of  energy  between  the  resonators  and 
the  ordinary  particles.'7 

These  extracts  sufficiently  indicate  the  nature  of  the  prob- 
lem that  has  arisen  in  prosecuting  this  research.  We  cannot 
practically  compute  from  the  electromagnetic  radiation  to  the 
temperature  of  the  ponderable  matter  in  the  assumed  ideal 
enclosure,  but  we  must  proceed  from  the  temperature  of  the 
free  gases  in  an  atmosphere  back  to  the  radiation  energy  which 
originated  it.  We  cannot  isolate  a  given  volume  of  the  aether 
having  a  certain  amount  of  radiant  agitation  in  space,  some- 
where between  the  sun  and  the  earth,  place  within  it  a  pon- 
derable mass  of  gas,  and  note  its  temperature.  Furthermore, 
it  is  not  necessary  to  have  resort  to  this  ideal  enclosing  vessel, 
which  is  useful  only  for  analytical  purposes,  because  the  single 
fact  which  is  essential  is  that  the  amount  of  radiant  energy  in 
the  unit  volume  should  remain  constant  for  the  temperature  T. 


COEFFICIENTS  IN  STEFAN  AND  WIEN-PLANCK  FORMULAS      125 

This  constancy  of  radiant  energy  per  volume  can  be  secured 
by  opening  the  ends  of  the  vessel  along  the  path  of  the  solar 
radiation,  and  so  allow  the  volume  energy  of  radiation  to  flow 
through  it  with  the  velocity  of  light.  In  this  way  all  unit 
spaces  surrounding  the  sun  are  filled  with  a  given  amount  of 
energy  fixed  proportionally  to  its  distance  from  the  sun.  The 
earth's  atmosphere  is  immersed  in  such  a  flowing  radiation, 
whose  intensity  per  unit  volume  within  the  atmosphere  is 

equivalent  to  cr  =  —  in  a  given  volume  per  degree3.     Since 

our  computations  are  strictly  thermodynamic,  being  based  upon 
the  ponderable  gas  temperatures  of  the  successive  strata,  we 
obtain  at  once  the  o-  of  the  Stefan,  or  quasi-Stefan  law,  and 
not  the  primitive  energy  of  the  aether  at  rest  in  a  vacuum 
outside  the  atmosphere.  The  computations  showed  immediately 
that  in  solving  the  expression, 

ff,-  U,  a 

^ 


the  c  is  to  be  identified  with  the  coefficient  cr,  and  this  is  the 
method  which  we  have  pursued.  Having  KIQ  and  T  it  was  neces- 
sary to  solve  the  equation  for  two  unknown  quantities  (c  and  a), 
as  has  been  done  by  the  method  of  pairs  (log  c,  a)  .  This  value 
of  c  is  so  closely  identical  with  the  Stefan  <r,  in  70  =  *  T*,  that 
the  problem  of  the  relations  between  c  and  <r  become  of  primary 
interest.  The  outcome  of  this  method  is  such  as  clearly  to 
justify  the  procedure,  and  to  bring  forward  many  important 
facts  which  will  be  of  value  in  solving  this  great  physical  prob- 
lem of  the  mechanism  of  the  transformation  of  the  radiant 
energy  of  the  aather  into  ponderable  kinetic  energy  at  the 
temperature  T,  or  the  reverse.  It  has  been  shown  that  the 
value  of  log  a  is  discontinuous  near  the  bottom  of  the  isothermal 
layers  of  the  gases  of  the  sun,  but  that  the  mean  logarithmic 
value  agrees  closely  with  the  Kurlbaum  value  of  this  coefficient. 
Further  examples  of  this  relation  will  be  given  in  this  chapter. 
(1)  The  model  for  the  analysis  of  radiant  electromagnetic 
energy  consists  in  enclosing  it  within  a  certain  volume  by  means 


126  A  TREATISE   ON  THE   SUN^S  RADIATION 

of  perfectly  reflecting  mirror  walls,  or  in  an  absorbing  enclosure 
at  the  fixed  temperature  T.  Such  an  amount  is  indestructible. 
(2)  The  same  volume  density  of  radiant  energy  can  be  main- 
tained at  the  same  point  in  the  pure  aether  of  space  by  the 
continuous  flux  of  the  constant  radiation  from  the  sun  outwards, 


8un 


FIG  20.  The  Several  Stages  in  the  Theory  of  the  Transformation  of  Radiant 
Energy  into  Molecular  Kinetic  Energy  at  the  Stationary  Temperature  T. 

after  enclosing  walls  have  been  removed.  The  amount  of  flux 
through  this  volume  (2)  per  second  is  3  X  1010  times  the  amount 
in  the  first  aether  volume  enclosed  in  (1). 

(3)  The  amount  arriving  at  any  equivalent  volume  (3)  in 
the  earth's  gaseous  atmosphere  per  second  is  equal  to  the  pure 
aether  electromagnetic  energy  which  is  contained  in  a  volume 
3  X  1010  cm.  long  X  1  cm.2  =  3  X  1010  cm.3 

(4)  If  this  amount  of  energy  is  instantaneously  enclosed  in 
1  cm.3  of  the  gaseous  atmosphere  of  the  earth,  in  the*  course  of 
time   this   electromagnetic   energy  of   the  aether  will  be  con- 
verted into  the  kinetic  energy  of  the  molecular  motion  of  the 
gas  at  a  fixed  equivalent  temperature  T.    The  electromagnetic 
energy  in  the  aether  possesses  no  temperature,  and  it  is  of  an 
entirely  different  physical  order  from  the  temperature  of  the 
gas,  but  the  two  kinds  of  energy  are  exactly  equivalent.     The 
dimensions  of  the  two  amounts  of  energy  are  always  equivalent 
to  the  kinetic  energy  of  the  gas  per  volume, 

rMJJ_    1  "]    kinetic  energy 
LT2    'X*J'        volume 

and  this  is  not  to  be  confused  with  the  rate  at  which  the  aether 
energy  flows  through  the  boundary  of  the  gaseous  volume  (4), 


\  J^LJ^      L    -  ~1   kinetic  energy 
L     72    '  D'  rj'    area,  second 


COEFFICIENTS  IN   STEFAN  AND  WIEN-PLANCK  FORMULAS   127 

The  reason  for  this  is  that  the  stationary  temperature  T  has  been 
acquired  only  after  the  lapse  of  time,  which  is  necessary  for  a 
certain  amount  of  electromagnetic  energy  in  this  volume  to  go 
through  the  physical  transformations.  The  flux  of  energy  per 
second  through  a  surface  of  the  volume,  as  in  case  (2)  of  the 
pure  aether,  merely  serves  to  maintain  the  fixed  amount  of  radi- 
ant energy  which  can  ultimately  generate  the  gaseous  tem- 
perature r.  It  is  easy  to  perceive  that  the  flux,  velocity  per 
second,  in  one  direction  through  the  aether,  from  the  sun  to 
the  earth,  is  broken  up  into  the  innumerable,  irregular,  direc- 
tional velocities  pertaining  to  the  heat  motions  of  the  mole- 
cules which  maintain  T,  so  that  we  cannot  utilize  the  flux- 
equations,  but  only  the  volume-equations,  in  this  connection. 
Hence,  the  values  of, 

ft-  ft  Ui- 


—    VQ 


Ta 

—  ff  * 


are   all  for  volume  energy   in    the    gases,    and    a  =    —  is  a 

c 

volume-energy  at  one  point  of  a  homogeneous  column  which 
is  c  =  3  X  1010  cm.  long.  It  is,  also,  the  pure  aether  energy 
within  the  gaseous  volume  without  any  concentration,  that  is, 
it  becomes  equivalent  to  the  true  electromagnetic  radiant 
energy  of  the  pure  aether. 

The  Coefficients  in  the  Wien-Planck  Formula  of  Spectrum 
Radiation 


(91)  /0 


1 


= 


^A         X2    *-"       \  2  "  \2  ^3  h  v 

C  A  A  A    (/  T-^, 


This  formula  is  constructed  on  the  basis  of  simple  black 
body  radiation,  which  is  adiabatically  confined  in  the  perfectly 
reflecting  mirror  walls  of  the  enclosing  vessel.  This  would 
correspond  to  the  pure  radiation  outside  the  vanishing  plane 
of  an  atmosphere,  but  in  the  passage  of  such  radiation  through 
a  non-adiabatic  atmosphere  the  distribution  becomes  different, 
and  it  has  been  the  purpose  of  this  research  to  discover  some 


128  A   TREATISE  ON  THE   SUN'S  RADIATION 

of  the  facts  which  are  concerned  with  the  generation,  propaga- 
tion, and  dissipation  of  the  solar  radiant  energy  in  the  atmos- 
pheres of  the  sun  and  of  the  earth.  The  Wien-Planck  formula 
of  distribution  is  closely  related  to  the  thermodynamic  condi- 
tions by  the  law  of  conservation, 

(114)  r10  (Si  -  So)  =  (ft  -  Co)  =  (wl  -  w0)  +  (tfi  -  Uo)  = 

Pio  fa  -  Vo)  +  (Z/i  ~  Uo). 

When  the  term  for  work,  (Wi  -  Wo)  =  PIO  fa  -  VQ)  =  0,  so 
that  there  is  no  volume  expansion  to  take  into  consideration, 
this  reduces  in  the  aether  to, 


(115) 


On  entering  an  atmosphere,  the  term  (Wi  —  Wo)  = 
Pio  fa  —  VQ)  is  valid,  changing  the  adiabatic  conditions  outside 
the  gaseous  envelope  into  the  non-adiabatic  conditions  within 
the  atmosphere,  these  prevailing  down  to  the  deep  adiabatic 
layers,  where  (Wi  —  Wo)  =  (Ui  —  Z70),  and  true  adiabatic  con- 
ditions are  restored.  In  the  non-adiabatic  layers, 

,ii,rt      ^-^     Qi-Qo  TW  (^  -  So)    . 

(lib)          -  =  --  /io  =  -  :  --  •  -LIQ* 

Vl  —  ^0  Vl  —  VQ  1)i  —  VQ 

Since  the  gas  coefficient  R  becomes  a  variable  in  the  non- 
adiabatic  strata,  it  follows  that  all  the  quantities  depending 
upon  R  are,  also,  variable  coefficients,  as,  for  example, 

(117)  General  gas  coefficient,  K  =  m  R. 

(118)  Boltzmann's  entropy  coefficient,    ^  =  —  =  ~^~. 

(119)  Planck's  Wirkungsquan  turn,  h  —  —  (-      —  J 

(120)  Wien-Planck  coefficient,  ci  =  Sir  c  h. 


COEFFICIENTS   IN   STEFAN  AND   WIEN-PLANCK  FORMULAS   129 

(121)  Stefan's  coefficient,  a  =  —  ,  where 


<7  =  variable,  corresponding  with  <r  in 


(122)  Spatial  density,  u  =  a  T*. 

(123)  Radiation  intensity,  K  =  ^  .  a  T\ 

c  =  velocity  of  light. 
4 

(124)  Entropy  (specific),  s  =  —  a  T3. 

o 

(125)  Radiation  of  Entropy,  L  =  —  .  a  T3. 

The  entire  thermodynamics  of  radiation,  therefore,  must  be 
based  upon  a  series  of  non-adiabatic  variable  coefficients,  instead 
of  upon  a  set  of  adiabatic  constants,  as  has  been  assumed  in  pre- 
vious discussions.  Furthermore,  we  have  already  indicated  that 
radiation  originates  in  a  per  saltum  process  of  readjustment 
of  the  electronic  circulation;  in  a  sudden  transformation,  and 
not  in  a  thermodynamic  change  of  the  kinetic  energy  of  the 
circulation  of  the  atoms  and  the  molecules.  It  seems  that  the 
radiation  originates  in  a  radioactive  process,  and  that  this 
radiant  energy  is  depleted  by  a  series  of  thermodynamic  proc- 
esses involving  absorption,  with  temperature  and  other  depen- 
dent changes,  as  well  as  scattering  by  reflection  without  tem- 
perature variations.  We  shall  proceed  to  collect  the  results  of 
the  terms  that  are  concerned  with  these  processes  of  thermo- 
dynamic distribution  in  the  Stefan  and  the  Wien-Planck  laws, 
now  computing  the  variable  coefficients  which  replace  the  adia- 
batic constants. 

The  Formulas  of  Computation 

The  computed  values   of    log  a  in  Kw  =  aTa  (M.  K.  S.) 

gram  calories 

have  been  transformed  into  -  —.  —   —  -  —  ;  —  —  by  the  conver- 

centimeter2  minute     J 


130  A   TREATISE    ON   THE    SUN'S   RADIATION 

10  X  60 
sion  factor    .  1QK1  ..  1Ay  =  0.000014336,  log  [-  5.15644].     We 

4.1oOl  /\  1U 


shall  now  proceed  to  reach  the  identical  results  by  an  entirely 
different  method  of  computation,  namely,  by  using  the  formulas 
of  Table  1  in  the  following  order.  It  will  facilitate  the  interpre- 
tation of  the  data  from  step  to  step,  by  thinking  of  the  energy 
in  terms  of  work  per  unit  volume,  which  is  the  kinetic  energy 
of  the  molecular  motion  per  unit  volume,  whose  dimensions 
should  be  written, 

ML2     1        mass  X  velocity  square        M 
T2    '  Z3  =  volume"  =  ~LT*' 

The  computations  proceed  in  order: 

O  I—  -\/r    7"  2         1—j 

(126)  H  =  —  Pj  |_—  pf~  *  Jt\t  the  molecular  kinetic  energy. 

(127)  J7  =  C2;pr,--=2,  the  total  inner  energy. 


Generally,  U  =  H  +  /,  the  molecular  \inetic  energy  plus 
the  inter-atomic  and  inter-molecular  energies  /.  In  the  mona- 
tomic  molecules  U  =  H  and  /  =  0.  Our  computations  have 
been  limited  to  monatomic  gases  on  account  of  the*  difficulty 
of  determining  J  in  the  sun.  We  have,  however,  carried  out 
this  work  for  hydrogen,  H2  =  2.00  diatomic,  because  this  gas 
vanishes  at  25000  kilometers  above  the  photosphere,  so  that 
Hi  =  1.00  monatomic  is  not  applicable.  For  H2  =  2,00  the 

value  of  jj  =  0.6094.     Hence,  /  =  U  -  H  =  0.3906  U.     This 

was  derived  by  assuming  the  ratio  of  the  specific  heats  K  = 
Cp/Cv  =  1.4063  for  H2  =  2.00,  the  same  as  for  atmospheric  air. 
This  assumption  seems  to  have  been  justified  for  hydrogen, 
H2  =  2.00,  but  the  difficulty  of  determining  K  =  Cp/Cv  for 
more  complex  solar  gases  is  such  that  this  part  of  the  research 
is  still  under  consideration. 

As  these  thermodynamic  quantities  are  based  upon  the 
velocity  of  the  molecular  motions,  the  next  pair  of  values  is 
necessarily, 


COEFFICIENTS   IN   STEFAN  AND   WIEN-PLANCK  FORMULAS    131 

(128)  <f  =  3R  T=  3  P  v,  the  mean  square  velocity,  which  is 
three  times  the  volume  kinetic  energy  for  the  unit  m. 

8  — 

(129)  72  =  —  <f   =  the    arithmetical    mean    velocity    square, 

O  7T 

used  in  computing  the  Maxwell  free  path  length  lmax,  and 
the  number  of  collisions  per  second  v.  Up  to  this  stage  the 
computations  have  employed  the  specific  kinetic  energy  formula, 

P  v  =  R  T,  or  P  =  p  R  T, 
where  m  =  1,  and  the  density  p  is  for  the  specific  volume  v  —  -. 

We  must  now  use  the  molecular  volume,  V  =  m  v  =  — ,  because 

P 

we  require  the  actual  amount  of  energy  in  the  operation.  The 
factor  m  readily  eliminates  from  the  group  of  formulas  here- 
tofore employed,  so  that  it  was  not  convenient  to  use  it.  We, 
therefore,  take  the  general  formulas, 

(130)  Pvm  =  mRT,     P  V  =  K  T,    P  =  -~?  =  p  R  T. 

The  value  K  thus  computed  is  required  in  determining  the 
Boltzmann  factor  &  =  — . 

r     L?    "| 

(131)  K  =  m  R,  the  general  thermal  efficiency,  I  ™~i —  I 

TJ        r—    I     __ 

(132)  n  =  — ,     —  1,  the  number  of  H-atoms  per  unit  volume. 

JiiQ    LJL  _J 

(133)  E0  =  - —  =  pr  w  w2,     —  h  the  mean  kinetic  energy  of 

2i  tl/        £  *~  J.     -J 

one  H-atom.      It  is  a  constant,  whose  value  is  5.5718  X  10"1 
[  -  14.74599]  in    the  earth's  atmosphere,  and  4.3769  X  10~n 
[  —  11.64117]  in  the  sun's  atmosphere. 

transformation  of  E0  (earth)  X  72  =  EQ  (sun) 
M.  K.  S.  C.  G.  S. 

Earth,  5.5718X10"21  [-21.74599],  5.5718 X  10~14  [- 14.74599] 

M. 


132  A  TREATISE   ON   THE   SUN^S  RADIATION 

Sun,  4.3769  X  10~18  [-  18.64117],  4.3769  X  lO"11  [-  11.64117] 


J 


The  transformation  is  E0  (earth)  X  7*  =  E0  (sun). 
72  =  (28.028)2  =  785.56,  [2.89518]. 


/«\     *  °  .   ^ 

(134)  u2  =  —  —  -  =  —  —  —  =  -  ,  is  the  mean  square 

velocity  of  the  single  molecule,  and  hence  the  mean  kinetic 
energy  of  each  molecule  in  the  unit  mass,  the  gram.  This  is 
not  a  constant  for  all  gases  because  m  is  in  the  denominator. 
Having  computed  u2  for  any  gas  under  terrestrial  conditions, 
u2  (earth)  72  =  u2  (sun);  also,  u2  (M.  K.  S.)  X  107  =  u2  (C.  G.  S.). 


(135)  N  ^=  Vn  =-=  -     ,     _,  the  number  of  mole- 

P  k       mH    LMJ 

cules  per  unit  mass,  the  gram  in  (C.  G.  S.).  This  is  supposed  to 
be  a  universal  constant,  the  reciprocal  of  the  mass  of  the  hydrogen 

atom,    —  =  N.  but  the  computations  in  non-adiabatic  atmos- 
t»fr 

pheres  show  that  there  is  some  variability.  Since  R  is  variable, 
K  =  m  R  is  variable,  and  this  variability  proceeds  throughout 
the  entire  system,  changing  constants  into  variable  coefficients. 
Next  we  compute, 

(136)  k  =  ft  =  -g-^-  =  R  m  X  mu  [j^  —    >  :S  the  Boltzmann 


entropy  coefficient. 

(137)  k  =  *L    i,  -  r,  erg  sec.,  the  Planck  Wir- 


kungsquantum.      a  =  1.0823   as   developed   by    Planck,   "  Die 
Theorie  der  Warmestrahlung,"  page  165, 

-      «=l+^+y4+^4  +..  .  =  1.0823. 

(138)  a  =  -  -  as  determined  from  K^  =  a  Ta. 
c 

Since  k  is  a  variable  coefficient  in  non-adiabatic  atmospheres, 
it  follows  that  h  is,  also,  a  variable  coefficient. 


COEFFICIENTS  IN  STEFAN  AND  WIEN-PLANCK  FORMULAS    133 


Wien-Planck 
coefficients. 


(139)  a  =  Sir  c  h,  L-JT-.  £J>  erg  X  cm. 

(140)  c2  =  -T-,  [L  .  deg.],  cm.  deg. 

(141)  a  =  ^H  -1FT  '  Ts '  ^il'  .JfL^   Stefan  coeffi- 


cients. 

c        rJf  L2     1     L      1  M     "1       gram 

(142)     ^-^[-r.-.-^-.^—^^,1^—-t 


deg." 
erg 


cm.2  sec.  deg.4 
(143)  A  =  4.1851  X  107,      "^r'  ergs,  mechanical  equivalent 

of  heat  in  small  calories  per  sec.2 

60  ff      gram  calories  , 

—  •  r—  =  -  -  ;  —  :  -  9  the  measure  of  the  kinetic  energy  oi 
A  cm.2  mm. 

molecular  motion  in  calories  of  heat.  Since  all  the  terms  of 
the  equations  of  condition  represent  kinetic  energy  of  molecular 

motion,   [_  I  >  in  one  form  or  another,  and  since  A  repre- 

sents a  definite  amount  of  this  kinetic  energy,  or  work  done 
per  unit  volume,  it  follows  that  they  can  all  be  reduced  to  the 
corresponding  number  of  calories.  The  computations  prove 

r*f\ 

that  —  -r-  X  10  derived  directly  from  the  (log  c,  a)  set  of  pairs 
A. 

(M.  K.  S.)  through  the  factor  0.000014336,  is  exactly  the  same 
in  value  as  when  derived  through  the  general  formula, 


Indeed,  all  the  resulting  values  of  a  (M.  K.  S.)  have  been  checked 
by  this  independent  computation  in  the  (C.  G.  S.)  to  verify  the 
clerical  work  of  the  computations. 

The  following  tables  contain  the  data  of  the  several  elements, 
as  they  are  collected  together  for  intercomparisons,  and  for  the 
derivation  of  the  general  laws  which  underlie  the  solar  thermo- 


134  A  TREATISE   ON  THE   SUN?S   RADIATION 

dynamics.  This  fragmentary  collection,  apart  from  the  con- 
tinuity of  the  computations,  necessarily  deprives  the  data  of 
much  of  its  original  force.  The  number  of  points  found  in  the 
tables  is  about  one-third  of  the  number  for  which  the  computa- 
tions were  made.  They  are  quite  full  near  the  photosphere,  but 
in  the  high  and  the  low  levels  the  data  have  been  very  much 
diminished  in  the  amount  to  be  included  in  the  tables. 


The  Thermodynamics  of  Radiation  in  the  Solar  Atmospheres 

Q 

The  table  for  H  =  —  P  has  been  omitted  because  it  can  be 

LI 

immediately  computed  from  Table  9;  similarly,  the  table  for 
U  =  CvpT  =  His  omitted  because  our  compilations  are  for 
monatomic  gases,  except  hydrogen,  H2  =  2.00,  as  has  been 

/  8  \i 
stated;  also,  the  table  for  7  =  (^J2  q  =  0.92132  q   [9.96441] 

is  omitted.  The  computations  for  the  Maxwell  mean  free  path 
length  are  not  computed  on  account  of  the  difficulty  of  deter- 

77 

mining  the  coefficient  v\  in  lmax  =     Qno^7 — ' '  even  ^  tne  coen^- 

U.oUyoi  p  7 

cient  0.30967  is  still  available.     Consequently,  the  computation 

7 
for  the  number  of  collisions  per  second,  v  =  1      ,  must  be  de- 

tmax 

ferred. 

Table  31  contains  the  square  root  of  the  mean  square- velocity 
q=  V  22.     An  examination  of  the  results  along  the  common 
plane  of  the  photosphere  for  the  different  elements  gives  the 
general  value, 

q  =,54289  /  V~^".     Hence,  \  m  f  =  1.4736  X  109  [9.16838]. 
The  universal  mean  kinetic  energy  is  equivalent  to, 

(145)  |m^  =  |pwz,=  |w^r=|pF=|^r=1.4736Xl09, 

which  is  the  same  as  for  monatomic  hydrogen.  This  conforms 
to  the  Clausius  law  of  the  constant  mean  kinetic  energy  at 
the  photosphere. 


COEFFICIENTS  IN  STEFAN  AND  WIEN-PLANCK  FORMULAS  135 


i-i  oo«ooo  t-co  10 
eo  TP T-ITJIOOO  eo 


O  T—i   < 

loojcoioc-  eo  ooeoi 

OO    IOOSOJ1OOO    CXI  ?OOTt<< 

rH  eoeoTj«Tj<-^  10  io5o«o< 


U5MCO   OOO^*COT-I   rH  eOCONOUJ 

eoi-io  oscoiot-Oi  »H  wot-«oi>  uso 

t-  •«*  W    t-OCOt-CXI    OS  «OlOTj<U3t>    OU3 

•en  1-1  • 


OONt-COt-    Ot-t-lON   O   t-OOC<l(M«> 

C»CT5U500r-l    t-T-tO^J<eO    t- 

ooeooo-^oo  oot>^uDio  10 


i-iT)<  ot-meoco  t- 
i-Heo  oot-uioso  eo 

OO5    OCO«£>OOi-(    Oi 


T)<  iH  CO  i-l  Tj<    OJt>OlfllO   rHO>?OCOO    t-    COOt-Tjli-( 
COVOOO(MCO    I0p«>i-lt-    U30p<N«00    CO    t-  T-H  «t  OO  <N 
OOi-H    T-H    i-l  CM  O3  C<I  CO 


t-OB£S    Wt-O^t-    OJOJOO^t    rH 
COi-INOl    OSOSOOO    OOrHi-li-(    i-H    . 

rHCSieo  eoeo^TCTt  rj< Tf rj< TJ< TJ«  T*  TI* ^ -«f  -^ •<*  -^< ^ ^ -^ ••* 


|  t-  t-  O  »H 

00  00  00  -^ 

4  co  oo  co  oo 


I  I  I 


Ig^s 
^Tw 


w  s 

'o      I 

s 

1 


2^ 


SIS    § 


a; 

N 


136 


te'u 


§^|s 

I  ' 

y  ^;  |^» 

« ^ 


^3 


A' TREATISE   ON  THE   SUN'S   RADIATION 

o 


X  XX        X 

T)<  os  os  t-  mc^ 

os  osooo  t>  t- 

t-  O5CO  O500O 

O  OOrH  COCDrH 


00  O  t-  N  CO 

is!!! 

CO  I*  rH  C<I  CO 


I  !--!- 

x   x"  "  x" 

CO        lOCOOcOOS 


OSOSO  •<*  00 

coos      TJI      TJ<  oo  10  eq  T}< 

OOCJCOrHrH        OS        COCOt-MC- 


fl 


rH         COlOOOrHrH         N 


XXX  X 

o  t-  m      •**•*!<  t- TJ<  o> 


SIO        COOOCOt-U5  rHO 

10         COrHCOOt-  rHOO 

^HTJIC-       rH        CO  CO  OJ  00  00  COO 

N*  CO  CO  Tjl  10  rH  l> 


sill-  -  S-  — 

xxxx"  "  x"  " 

t-NOOOCD  Nt-mO< 

OSCJ^C^rH  T^t^CO^OC 

COWOOO  WrHN^C 


O  t-CO< 

U50JO( 


X     X 

U5  COrHOS  t> 
O  CO  CO  CD  CO 

©CO  05  rH  05 


X        X 


)  OSO  rH         rH 


t-  cowl1  -^      OS' 


|o     o,   .  . 
XX     X" 

OrH         WOOOJ< 


-I- 1 

"  x"  x 

t-  CO  O  U5 

t»  N  om 


>rHOco  •*  mt-oooos  comooin'^' 

looeooo  co  os>oc<iooio  corjieot-eo 

)Oco»o  oo  ococoooi-i  us  co  rH  o  m 

ic<ic<i<N  N  W  M' N' W  <N  (NJCvJCOCOCO        t>N«3rH 


00 

si 

I  t- 


si 

?2 


XXXX  X 

OOt~iO-^<C<l  OJCOCO 

lOTjtt-OOrH  OCDt- 

OJOOOJ  rH  TH  OO 


3 

rH  rH  CO  N  CO 


rHN         OOOrHNCO          Ifi         CO  t- OS  O  rH          00  (N  ^  ( 

OOOS        OSOOOO        O        OOOrHrH        rHCOrfCOOO        OOrHCOTj«(NOJO 

rH  c<i  c<i  ca  c<i     N     e<i  <N IN  c<l  c<J     c<JN<MC<ic<J     co' i»' rn  N -^  co  r-J 


X  X 

00  O  CD  CO  rji  t^  Ifl 

CQ  s^  csi  to  oo  co  10 

00  rH  CO  " 


111 

xxx 


x 


CO"300rHU3        00 
O  CD  <N  OS  iO         rH 


^sssl  § 


-*OJ(N 
W  t>  IO 

cj  co  o  co  oj  oo 

00  00  (N  i-t  t- CO  t- 

N  CO  10  t-'  OJ  rH  rH 


.h  '  oo 


§0000 
OOCOTf  (N 


1     1     1 


i 


OOOOO  OO 

o  o  o  oo  o  o 

O  O  OOOOO 

eg  Tj«  CD  oo  o  cxi  •<* 

1  1  1  1777 


COEFFICIENTS   IN   STEFAN  AND   WIEN-PLANCK  FORMULAS    137 

The  data  for  the  planes  of  the  radiation,  ZR,  give  a  similar 
result,  but  with  a  different  constant,  q  =  71241/  ^Jm.  Hence, 
J  m  q2  =  2.5376  X  109,  [9.40443]. 

Finally,  on  the  vanishing  planes  q  =  0  and  the  mean  kinetic 
energy  vanishes.  There  must,  therefore,  be  a  complex  gradual 
diminution  of  the  general  mean  kinetic  energy  from  level  to 
level,  counting  from  the  inner  layers  to  the  vanishing  planes. 
The  values  in  the  lower  levels  can  be  computed  as  long  as  the 
gaseous  law  P  =  p  R  T,  or  P  V  =  K  T,  continues  to  represent 
the  natural  conditions,  that  is,  before  viscous  coefficients  become 
necessary. 

The  value  of  q  is  important  in  several  formulas. 

The  general  kinetic  energy,      PV  =  KT==~mq2. 

o 

The  specific  kinetic  energy,       Pv  =  RT  =  -q2. 

o 

The  hydrostatic  pressure,  P  =  p  RT  =  —  pq2. 

o 

3  1 

The  molecular  kinetic  energy,     H  =  -  P  =  -  p  q2. 

2  2 

By  Table  9  the  mean  pressure  along  the  layers  where  the 

black  radiation  originates  is  P  =  2593212  (M.  K.  S.). 
3 

Since  H  =  —  P  =  E0  n,  we  have  in  M.  K.  S.  for  n  =  8.8990  X  1023 
2 

H  =  3889800,  and  E0  n  =  3895000, 

which  checks  the  procedure  approximately,  especially  the  re- 
sults by  the  method  of  trials  upon  which  the  entire  series  of 
data  depends.  The  average  number  of  molecules  per  cubic 
centimeter  on  the  plane  of  the  photosphere  is  2.1187  X  1017, 
about  one-fourth  the  number  on  the  radiation  levels.  The  num- 
ber diminishes  to  zero  on  the  vanishing  planes,  very  rapidly  in 
the  topmost  strata.  The  number  of  molecules  per  cu.  cm.  on 
the  plane  of  the  photosphere  is  about  one-hundredth  the  amount 
in  the  earth's  atmosphere  near  the  sea  level,  2.1187  X  1017  against 
2.7278  X  1019. 


138 


A  TREATISE   ON  THE   SUN  S  RADIATION 


03 

i 

. 

A 

k 

i 

ft!  ^ 

kJk 

IN  00 

£1 

woo 

W<N 

•SfS 

Ci       00  ^  CD  W  ^* 

a 

OSTHt»TH 

^  0}  TH  ^ 

oooi  t>oo,       -   »   »  .   -. 

O5         OSO  TH  COO5 
N       -*CDl>000 

(M 

TH 

10  o  mcD 

THNMIN 

/—  X 

C/5 

»O       iHCO^t  O  UO 
W         THCDTHOO  O 

TH     odaiO  10 

O5 

OOCOU300  O       O5 
t-rH  N  O5  W         ^ 

* 

0^ 

•«tf       OOOOOSOiH 

1 

N  IO  TH  N  T)<         CO*      -      -      - 

IO  t-O  CO  CO        t- 
i-(TH<N<NW       (N 

£ 

•^•THO      OiCDCDcoas 
ioooo      t-oocooas 

1 

OO  (N  lOCO  t-        OOO 

ta 
II   *    g 

N<D 

ira 

THTHTHTHrH         M  <N 

^       " 

»»ggg  gsssg?; 

§ 

gg|3g    .||| 

5  §  ^ 
£§^ 

°" 

CO  iH  CO  ^D  t"        O^  O5  O  O  'H 

TH 
TH 

COOCD^TH        M^^f*- 

I       " 

S88S3SS    SSES2§a 

s 

CD 

OO^CONIN     oooooe^ 

s 

S    ^    r  * 

M      Q        h»i 

^TH 

<Dt>OOO5O        TH  TH  C<1  (M  C<1 

CD 

CMCOCOCOCO     Tj<t-a>coco 

w 

§  , 

^      t> 

COT**        CDNOCD-^        WCD-^CDN 

M 

T}<CDCDC<|CD       OW^COOO 

NW(MM 

i  s 

^^ 

NCO      TH-^OTHOO      OO-^OCDCO 

COO       OONt-CMt-        OiH(MC<ICO 

1 

kOrHt-COO         iHt-t-THOO 
•^IC^SCDt-         OCDCOiHOO 
N<N(NIN(N        COCO^lOiO 

1H|S    :    : 

H    ii 

H        a 
S        % 
* 

^d 

i!II  iiiii  IIII! 

1 

CD  Ti<  CO  ^  O        CD  CJ  ^  O  O 

CDT^THCT>t>        lOt>O^THOO 

"*  W  O  «O  00  OO  OO 

ia  t-  Ti<  o  c<i  csi  cj 

*C>3lO'^GC'^'^^t< 
CO  TH  W  00  W)  *O  IO 

•  7 

THWNNWWCC, 

H    ^ 

rH 

OOCOt-NCO        OOOOW5        COTHOO<M 

1 

OOOOO        OOOOO 

coooTfoio      rj<cococDTH 

CDt-OiTHOl        OCDINOOU3 
COCOCO-«t^        U5CDOOO5TH 
(N<N(N(N(N        <N<NIN(MCO 

o  1000  m  TH  ^f  o 

Tf  lO  t-  O  CO  CO  CO 

M       " 

8 

W 
J 
PQ 

< 

:     :     :      :  :  :  >       :  :  :  : 

H 

•v!                '.    '.    '•    '• 

i  :  :  :           :  :  :  : 

o'  d  o'  o  c>     ooooo     d  o  d  d  o' 

OOOOO        OOOOO        OOOCD^d 
IOOIOO  "3       T-I 

N 

d 

1      I      1      |    rH        W  "31  SO  00  O 

1  1   1  1  1  if 

c^coooo^ 

COEFFICIENTS   IN   STEFAN  AND   WIEN-PLANCK  FORMULAS   139 


CO  |OQ 


V) 

W      Ei 


a  "I I 


s* 


X  XX 

"*  COCMt-t-rH  (O  r-ICReO 

US  ^O^COi-H  T-H  OINCJ5 

i-H  CO  CM  "<t -<t  •**  CO  OOCO^i 

US  COO^OSUS  CM  rH  CO  00 

CM  CD  i-H  i-l  i-H  CM  CO 


O        O 


X     X 

t-       COO  CO' 


XXX 

osusos      us- 


l$%      5 


USOSrH        CMCvICMCMOS       CO        CO  CO  •**  •**  ^        t- i-H 


Ill-s 

xxx"  x 


X 

eooiocoo      cooi-ioo'O      us      eoosT}«t--o      co-^cMt-i 
o  co  -^  t^-  CD      CM  us  oo  IH  co      I-H      t*- os  IH  os  os      cot— usos< 

COO5USOOOO        rfUSCOOO''*       rH        CM^COt-OS        eOCDC<l<D< 

ojuscoos'i-I     CM"  CM  CM  CM' CM     os'     co' co  co' os' 03     w'oo'r-Ir-K 


T-HT-IOOCMCM     o     coususco< 

^OCD^CM       TH        ScM^b-i 
OUSOSTJ«0>        rjl        OS -*  OS  T£  < 


r-lTHCMCMCM       OS  CO  CO  OS  CO 


eooscoosos     eo-*uscot> 


II     I:   =  ,  -.       ,,,••.'.       =       =  =  =  =  =       =  =  =  =  =       =  =  1  = 


XX     X 

lr-1         USOOSOOOO 


CM  CO  i-l 
00  OS  rH 


CMOO-^OCO       CM       OO^i-Ht-CO 

oo  o  c<i -^  us     t-     OOOCMOSUS 

THtMNCMCM       CM       CM03C003CO 


X 

ooaoo 


eoeocoosco      co      cooseoosos      eoeocoos-* 


xxxx 


US^HINt-IUS         COOCOCOOS 
CMINCMrHCO        USCOCOt-t> 

•*3<  eo  co  1-1  CM     coeoeoosco 


(M  oo  in  cvios 

OS  OS  COCO  CO 


rfCOCOCOCO       COCOCMUSi-l  OSTj<OOOO( 

030t>-*i-H         COOO^HUirH  t>  OS  FH  t>  rH  ( 

OSOOOUSOS  CMCOeOrHOl 

OSOOrHCM  COU3CPOseO< 


0000000005 

eo'  os  eo'  co  co'      eo  •*'  TJ  TH'  n< 


US  (0  t>  Os  iH  r- 


1:1: 

x"  x* 


SrH         COO< 
OS        05rH( 


eot-i-HrHCM     eocoeoeoco 


CM  CM  CM  <M  CM 

os'  eo'  os  co'  co' 


^  co 

CO  CM 

oo  t- 


eoososeoos     eoeoeoosco 


oooo 

(M^CDOO 

1  1  1  1 


1    1     1 


' 


1     1     1 


IS 

XX 


IS 

000 

co'io 


ecus' 


140  A   TREATISE   ON  THE   SUN'S   RADIATION 

The  values  of  K  =  m  R  in  Table  33  are  in  the  (M.  K.  S.) 
system,  to  be  converted  into  (C.  G.  S.)  by  the  factor  104.  There 
are  three  principal  constant  values,  KA  =  282628  at  the  top 
of  the  adiabatic  region;  KR  =  221672  in  the  strata  where  the 
black  body  radiation  originates;  and  KP  =  127704  on  the 
plane  of  the  photosphere,  the  decrease  in  the  value  of  K  con- 
tinuing to  zero  on  the  vanishing  planes.  The  variability  of 
K  =  m  R  necessarily  follows  the  variability  of  R.  It  is  a  con- 
stant in  the  adiabatic  region,  but  falls  in  value  in  the  isothermal 
and  the  non-adiabatic  regions  till  it  vanishes.  The  vertical 
decrease  in  K  is  slow  for  hydrogen,  and  it  is  very  rapid  for  mer- 
cury. The  family  of  curves  can  be  plotted,  using  the  mean 
points,  and  their  relations  to  the  height  will  be  seen  to  be 
hyperbolic.  Since  K  is  variable  it  is  probable  that  K  =  k  N 
are  all  variables. 

The  number  of  molecules  in  the  unit  mass,  here  the  gram 
molecule,  is  again  constant  in  certain  layers,  but  these  are 
slowly  diminishing  with  the  height.  At  the  top  of  the  adiabatic 
layers  NA  =  6.9210  X  1023;  at  the  radiation  layer  NR  = 
5.8270  X  1023;  at  the  photosphere  Np  =  3.2929  X  1023. 

We  now  encounter  a  difficult  physical  problem. 

(146)  NE0  =  ^rPV=  ^KT  =  ^-mRT  =  \rntf  is  mutually 
&  L  A  £ 

satisfied  for  each  gas  at  a  given  level;  the  several  gases  have 
certain  common  values  as  in  Table  34,  but  changing  from  one 
level  to  another,  it  follows  that  N,  the  number  of  molecules  per 
unit  mass,  and  EQ,  the  kinetic  energy  of  one  molecule,  £0  = 

3  P      1  .  3  mRT      3  KT  1  .     .     .       .        . 

—  —  =— •  mw2=  —  — ^ — =  TT-^-  cannot  both  be  fixed  universal 

2t  n       2i  Z      1\  Z    1\ 

constants  at  the  same  time.  It  has  been  commonly  assumed  in 
physics  that  the  kinetic  energy  of  each  molecule  of  a  gas  in 
collision  to  produce  the  pressure  P  is  an  indestructible  quantity, 
primitive  to  the  constitution  of  matter  itself.  It  is  the  force 
that  maintains  constant  the  structure  of  material  bodies,  what- 
ever its  nature.  It  is  of  the  same  primitive  character  as  gravi- 
tation, or  the  electromagnetic  forces,  and  on  this  account  E0 
has  been  retained  a  constant  while  N  becomes  a  variable.  It 


COEFFICIENTS   IN   STEFAN  AND   WIEN-PLANCK  FORMULAS   141 


0) 

i 

2         22 

2.5507X10" 

2 

X 

2:::   2 
X            X 

-    5  —  -  - 

o 

0 

<£>  lO  COO  (N 

t~       Oi  t*  t>  CO  Ci 

^io       § 

1  d      1 

0 

0 

«« 

X 

X 



oS     o 

^ 

m 

0 

COCXN^CO 
CO  OO  lOOJ 

kO       t*OCO  «O  O 

»o                                                 ^ 

§ 

00  CO        r^ 

Sift 

22: 

xx" 

-  2-  -  - 
"  x"  " 

:       :  :  :  :  5 

oseo                                          *• 

og    „ 

t-co  to 

(MOO  CO 

cooi  -^  o  co 
t-o  I-H  oico 

^      Jow^Soo 

10       CSCOOOINCO 

Tf  CO 

1%  " 

000 

0 

S3 

•  xxx 

oo8§!2g 

X 

CJ  (M  (N  Csl  CO 
CO  <NOO  Tf  O 

lO       O  CONOOTji 

co      oco  co  o>  co 

w  coco  co  oo     10 

ss  § 



n.     **.    .    .^    . 

1  ci      1 

CO 

0             0 

02 

X        X 

CO  COO  CO  M 

iiill 

CO  CO  •<*  T!<  •>* 

CO       UiiftiOlOlO 

CD  Oi  ff>  CO  CO        OO                                > 

oS     § 

IOCO       O 
t-10       rH 

tfi 

22    2— 
xx   x"  " 

»:,» 

i  :§§8^ 

OOioot-co      co  oj  m  o 

C<J  ^H  rH  r-l  i-H         W  t-  05  CO 

III 

-5 

|o&_  | 

xxx"  x 

1C  O  rH  O^  CO          'Tf  t—  O  CO  CO 

Ilili 

N      looo  IN  m  oo 

O>       0>O>  OOO 

-*  co  oo  o  I-H      in  ^  c\i  m  1-1  oi  oo 

COIOOOINIO        W  IM  O  CO  t>  t-  00 

Tf  10     «o 

'  N         ' 

0                   0 

>-5 

X            X 

OOU5NOJCO 

CO       OOO  »-H  CO  lO 

lOOSCOt-rH        O>  00  CO  t-  CO  W  O 

•^*C<I^O5OO         >O  CO  O5  O4  CO  TJ"  (M 

•**  CO  00  C5  i-(         r-(  !M  CO  fO  in  O  00 

Td       ^ 

N 

§0000 
ooco-tM 

0       OOOOO 
Ol  ^  CO  OOO 

1  1  1  1^ 

!««  IliiiS! 

1           J 

I 

£*      ^V 
Nj  N."      ts^[  s 

142 


A  TREATISE   ON   THE   SUN  S   RADIATION 


I. 


CO05    TfOOO         tHO    CO»H    i-l       O       t-lOO    tOtO         00    TjtlOlOCO         T-I    N  ^J*    t-*O>O 

i— i  O  CM  O5  05     C3  i"™*  COO  O    05    05  05  OO  O5  O5     O  05  O5  05  O     O  OO  O  rH  05 

coco  eo'eo'eo*       coco  co'eo  co     co     co  co'eo  coco'       eo  cocoeoeo       co'  coco'  co'eo  co 


t-rjt 

O005 
CO  00 


Tt<       OS«£>0 


00       OOOO  t- 

CO        CO        CO  CO  CO 


•< 

Is 


00    t-       l>       000000    00t> 


ss- 

XX* 

10  CO  05 


^-(O05   0000 

too  co  t-  I-H 


005  t-io  eo  N  <o< 

05  tO  Tf  CO    O  OO  OOO505    05< 

0000  0000    00  t-  t-t~t-    t-00 

coco  co'eo  eo  co  eo'eo'eo'  co'eo' 


I    H 

x"  x* 


o     ooo 

t>        ClrHCO 
OO       OO  Gi  C5 


coco  coco  eo     co     coeoco  coco 


05  1-  to 

5  Tf  OS  U3 
COt>Tl< 


>^  \n     o     10010  oio      o  050>ioeo 

t^       t-       t~l>t~    t^t-         l>    00000000 

coco  coco  co     co     eococo  coco       co  cocoeoeo 


5S5     g§gS 


ii 

XX 

Is 

co't> 


x   x 

i-l05  to  O 


coco  eococo       coco  coco  eo     eo     cocoeo  coco       eo  eocoeoco      eo 


ooo  o 

O  N    CO 
CON    rH 


VOIO    IO1 

os o  o< 

t>00    OOt 


coco  eococo       coco  coco  co     eo     eoeoeo  coco       co  cocoeoeo       eo  coco 


lOCOr-l  r-l 

t>  to  eo  10 

-t  t-  tO  •<* 

to  iH  t-  •<* 


ooeoggg 

05  05    050500 


NCOCO   Tj<rj< 

O3  05  05    05  05 


OOCO&OTJICO       «w  coeoco       coco  coco  co     co     eococo  coco       eo  eocoeoco 


§3 


COEFFICIENTS  IN  STEFAN  AND  WIEN-PLANCK  FORMULAS   143 

seems  more  probable  that  the  number  of  molecules  in  the  gram, 
under  the  extreme  solar  conditions,  should  be  the  variable. 

However,  since  N  =  — ,  and  mH  =  —  is    the  mass  of  the 
MH  M 

hydrogen  atom,  it  follows  that  this  is  a  computed  variable  in  free 
atmospheres.  If  it  is  insisted  that  mH  is  to  be  invariable,  it 
follows  that  EQ  is  to  be  made  a  variable.  The  only  escape  from 
this  dilemma  is  that  the  simple  thermodynamic  laws  laid  at  the 
basis  of  the  computations,  while  they  conform  to  the  analytical 
concepts  regarding  the  kinetic  theory  of  matter,  nevertheless 
are  too  simple  to  represent  the  facts  in  nature.  The  electron 
theory  of  matter  has  opened  up  such  extensive  possibilities  as 
to  the  physical  structures  of  atoms,  molecules,  and  their  rela- 
tions through  ionization  and  free  electrons  that  the  kinetic 
theory  of  thermodynamics  may  be  inadequate.  It  is  our  pur- 
pose to  present  the  facts  that  must  follow  from  the  variability 
of  R  in  free  atmospheres,  though  it  carries  with  it  a  revolu- 
tionary modification  of  many  old  and  current  hypotheses,  es- 
pecially regarding  the  origin  of  radiation  and  its  emission  and 
absorption  in  free  gaseous  atmospheres.  The  common  formulas 
are  based  upon  strictly  ideal  models,  as  black  radiation  in  a 
mirror  enclosure,  apart  from  influencing  ponderable  matter. 
It  is  necessary  to  define  the  existing  thermodynamic  conditions 
in  atmospheric  gases  before  proceeding  to  the  corresponding 
processes  in  electromagnetic  radiations. 

Table  35  contains  the  mean  kinetic  energy  \  m  q2,  and  sev- 
eral equivalent  forms.  It  has  a  constant  value  for  all  gases  on 
certain  levels,  at  the  top  of  the  adiabatic  strata  (J  m  q2)A  = 
3.0593  X  109;  on  the  radiation  layers  (%mq*)R  =  2.5507  X  109; 
on  the  plane  of  the  photosphere  (%mqz)p  =  1.4412  X  109;  on 
the  vanishing  planes  the  kinetic  energy  is  zero.  There  is  a 
gradual  decrease  in  the  value  of  the  kinetic  energy  from  the 
deep  layers  beneath  the  photosphere  to  the  vanishing  planes 
at  whatever  height  they  may  occur. 

Returning  to  the  problem  of  the  coefficients,  it  is  seen  from 
the  general  equation  that  we  have, 

tM*\  T?       %m<?      f  PV  _HV  _H  _$KT  _ 
(117)  I*  -  --  -  —-  -----  - 


144  A   TREATISE   ON  THE   SUN'S   RADIATION 

tr 

If  K  and  N  are  universal  constants,  and  k  =  —,  E0  —  f  k  T, 

it  follows  that  the  Boltzmann  entropy  product  k  T  is,  also,  a 
constant.  If  K  is  a  variable,  K  =  m  R,  as  it  must  be  if  R  is  a 
variable,  it  follows  that  k  is  a  variable  unless  N  varies  in  the 
same  ratio  as  K  =  m  R  varies  in  the  non-adiabatic  strata.  At 
the  time  of  making  the  computations  for  trial,  it  was  not  known 
what  any  of  the  relations  are  between  N  .n  .  H  .  EQ.  P  .  V  . 

TJ 

K .  T.    They  were  computed  in  the  order  K  =  mR0,  n  —  —, 

&o 

N  =  n  —  =  — —    and   checked  by  «2  =  ^;.     It  is  perfectly 
p         EQ  p  N 

evident  that  if  there  results  a  value  of  N  inconsistent  with  N  = 

— ,  where  mH  is  the  mass  of  the  hydrogen  atom,  computed  for 
mu 

EQ  =  constant,  log  E0  =  —  18.64117,  it  follows  that  N  or  EQ 
must  xme  or  both  be  variables.  The  former  as  a  variable  im- 
plies that  the  structure  of  the  mass  of  the  gram-molecule  depends 
upon  other  forces  than  the  mere  summation  of  independent 

N 

mass    molecules  2  mH,  such  as  are  being  so  widely  discussed 

in  the  electron  theory  of  matter.  The  latter  as  a  variable  admits 
that  the  molecular  kinetic  energy  of  individual  atoms  and 
molecules  is  variable,  and  that  E0  per  degree  has  other  conditions 

p  v 
affecting  it  than  the  temperature,  T  =  —^-.    The  theory  of  the 

free  path  lengths,  collisions,  invariability  of  the  primitive  kinetic 
energy,  are  all  involved.  The  choice  is  so  difficult  that  it  is 
thought  proper  to  present  the  problem  in  as  concrete  a  form 
as  possible.  It  may  be  noted  that  unless  N  diminishes  with 
the  height,  as  in  Table  34,  the  variability  of  the  Boltzmann 

TT 

entropy  coefficient  k  =  ^=  will  be  somewhat  greater  than  it  has 

been  found  to  be  in  Table  36;  and  the  Planck  Wirkungsquantum 
will  be  a  greater  variable  than  that  given  in  Table  38.  Further- 
more, the  values  of  Ci  in  Table  40,  c2  in  Table  42,  a  in  Table  44, 
will  be  much  wider  variables.  Compare  the  last  Chapter  VIII 
where  N  is  proved  to  be  variable. 


COEFFICIENTS   IN   STEFAN  AND  WIEN-PLANCK  FORMULAS    145 


|x" 

The  Boltzmann  entropy  coefficient,   k  =  -rz,   is   computed 

directly  from  K  and  N,  each  variables,  and  it  results  in  a  com- 
mon mean  value  throughout  the  isothermal  region,  including 
the  radiation  layer. 

k  =  3.8048  X  10~15  [log  k  =  -  15.58033]  erg/deg. 

The  common  value  for  terrestrial  atmospheric  air  at  the  sea 
level  is  approximately  1.3606 X 10"16  [-  16. 13374]  as  derived  in 
the  laboratory.  From  balloon  ascensions,  Bulletin  No.  4, 
O.  M.  A.,  we  collect  the  following  data: 

TABLE  37 

T^ 

k  =  - —  AS  DERIVED  FROM  BALLOON  ASCENSIONS. 


1911 

Uccle, 
June  9 

Uccle, 
Sept.  13 

Uccle, 
Nov.  9 

Means  up  to  60000  meters 

z  =90000. 
80000  . 
70000  . 
60000  . 
50000  . 
40000  . 
30000  . 
20000. 
10000. 
0000. 
The  labora 

to 

y 

3.7118X1 
9.2830X1 
5.3048 
3.7123 
3.0947 
2.1975 
1  .  6481 
1.6641 
1  .  6800 
1.3419 

LO-14 
LO-16 

4.1775X1 
3.2357 
2.2379 
1.6701 
1.6930 
1  .  6379 
1.2983 
1.3606X 

LO-16     3.7066X10-16 

*|jj 

1*1 

ess 

I0-lfl     -16.13374 

3.7139X1 
8.2530X1 
4  .  6427 
3  .  3766 
2.2374 
1  .  6656 
1  .  6904 
1  .  5895 
1.2699 

LO-14 

0_16 

3.  7125  X 
*14.0890X1 
*6.5164 
2.2789 
1  .  6962 
1  .  7246 
1  .  6443 
1.2832 
value.  .  .  . 

LO-14 

o-16 

The  ratio 


k  (sun)       38.0480X10-16 


k  (earth)       1.3606X10-16 


=  27.963,  nearly  y   =  28.028. 


*  Omitted  from  the  means. 


Table  38  gives  the  Planck  Wirkungsquantum  h,  which  in  the 
isothermal  layer  is  nearly  constant,  but  it  really  increases  slowly 
with  the  height.  Its  mean  value  is  h=  1.2113X  10~23  [-23.08325] 
in  these  strata,  generally  above  the  level  of  radiation  proper. 
Below  these  levels  we  have  the  following  values, 

ZR  -  9000  -  4500  -  2250  -  750  -  255  -  140  -  80  -  45. 

hA  8.6264  X  10~26  1.1608  X  10'24  6.4703  X  10"26 
7.6756  X  10"26  6.7790  X  10"26  8.0490  X  10"26 
8.0893  X  10"26  7.8370  X  10~26  7.6467  X  10"26. 

Omitting  hydrogen,  H2  =  2.00,  we  have  the  result: 

h2  above  the  radiation  layer  1.2113  X  10~23,  -  23.08325. 


hi  below 


,-26 


7.6467  X  10    ,  -  26.88347. 


146 


A   TREATISE   ON  THE   SUN  S   RADIATION 


I 

j 

i 

X 

o  o  co  -^eo       CM  T. 

OrH     rH  CO  m            t-  t- 

10  •<:)<  oo  in  co       os  a 

tMN    CMCMCM          tMO 

•4inc<iin     TJI      ososm 
•  osmo     m     rHt-os 
3  os  t-  m     e<>     TH  CM  I-H 

1    CM  CM    N        CM        CM    CM  CM 

II 

1 

CM 

CO  N 

coos 
os  t- 

co  co     t- 

OOi-H         t- 
t-C-       CO 

I 

I! 

X 

i—  1 

« 

1       f 

0         0. 

x    x" 

CO            rH  T- 
rH           CDO 
0         OOt- 
CO         t-C 

•                                        us      coe* 

1      ^ 
.     .      ,          .          0    00 

X  XX 

4    OS  t-     ^H         OS         t-    OO 

1     00  O     rH           00          CO     <N  t- 

>  Sco  TH      o     o  oooo 

ii 

I 

rH 

i 

1 

CO 
(M 
I 

03 

X 

CM         WC 

co       os« 

Tj<           rH  i- 

H    NOS    ON-*    COO 
3    ^  CO    i-l        CO        rj<    CMOO 
3    CO  C<I    00        •*        rH    OO  IO 

X 

coo 
cs  co 

00  t> 

t- 

co 

11  .. 

1 

1 

o 

$ 

XX 

t-oo-^<       os  t- 

COO^          CDU 

OOOOCM       mo 

OS  t—  O          ^  T 

•IOMOS        CD        t-OSi-H 

I  os  CD  co      o      oo  co  T}< 

1*    COCO    CO       CO        CM    CMCM 

X 

IS 

8.0490X1 

1 

CO* 

te 

XX  X 

i-H  CM    iH  OOOS          t-  t- 

o  co  03  w  TJ<       TH  c 

mt  i  in 

I! 

1 

X 

'    1 

.7790X10-26 
.1383  " 

.8862  " 
.1736  " 

.8926X10-27 

tMiH    TJIN^H           r4j. 

»H 

CD** 

CMCM        00 

1  

a 

0. 

us 

lull  li 

•  t—  t—  co      co      cDinm 

1    <MCM    CM        CM        CM    CM  <M 

I! 

i 

X 

03  CO 

X 

COOO        OS 

mt-     CD 
t-oo     t- 

11  ".. 

1 

i 

£- 

XX 

CD-*          ^,35    cOt-CO         COt 

w  t-       in  CD  OOOCM       cor- 

CO  00 

0 

cog 

X 

§8£    § 

X 

CO 

o 

|L 

osm       ri<eo  eococo       coe> 

3    COCO    CO       CO        CO    COCO 

coco 

CO 

NCM 

CMCM        CM 

CD' 

rH  t- 

111.  . 

1 

1.    I.. 

% 

XXX 

COCDOi-ltM          COOS    (M>000         OU 
lOOSOrfCO          O5CO    OONCD         TJ<  o 
OSCMC-COCO          inin    rfrJ<CO         COP 
T-HCOtMCOt-           ^H^H    ^HrH^H           r-lr 

m  i  in 

S3 

10 

in 

CM 

t-os 

S2 

rH  rH 

00  M        CM 

ss  § 

1.0028X1 

X     X 

00  ^C    ^  00  Ci 
iHO    OOCOrH 

s% 

I-.  -.  =  , 

X 

CDOOOOCO         CMCO    TjtTjirl*         OSC1 

•*  os  o  I-H  co       mcsj  oscoco       »HI- 
ooiOTj"int-       oooo  t-  1-  1-       t-t 

SCOOt—         Tj*         rHOSCO 
H     rH  rH     OOO     OSOS 

-t-t-t-        t-        t-CDCD 

coo 
os  os 

CO  CD 

t- 

co 

CO  CM 

•*  iH 

THO           t- 

oo  in     os 
m  m     eo 

t> 

0912  " 

0738  X  10-23  1 

6264X10-26) 
5400  " 
4750  '  1 

^NtMrH^H           ^rH    ***           r-Hr 

'.'.'.'.'.        '.'.'.'.'.        '. 

1    • 

ddddd      do  odd      dc 

OOOOO         OO    OOO         OO 
OOOOO         OOO    CO^CM         rH 

CMNr-1,-1 

N 

3    OO    OOO    OO 
0    COTl<    CM                   CM    TfCD 
1       1      1 

i 

2 

0H 

"17 

i 

11 

1 

'7  T 

o 
o 
o 

7 

O  O    OOO 

""777 

1 
§ 

COEFFICIENTS  IN  STEFAN  AND  WIEN-PLANCK  FORMULAS    147 


We  have  here  a  sudden  transition  in  the  value  of  h,  similar 
to  that  discussed  for  <r  under  Table  24,  by  the  method  of  the 
logarithmic  spiral, 

log  <72  for  the  final      point  ,—  9.81400. 
log  (70   "     "   middle      "      -  5.73690. 
log  ffi   "     "   initial       "      -  2.25450. 
The  distance  of  log  o-0  from  log  <n  =  0.461  of  the  total  dis- 
tance from  log  cri  to  log  <r2.     Similarly,  the  logarithmic  mean 
value  of  ho  can  be  obtained  from  h2  and  hi  by  applying  the  factor 
0.461  to  the  interval. 

log  hz  =  the  final      value,  —  23.08325,  approximately, 
log  hQ  =    "    middle      "      -  25.13167, 
log  hi  =    "    initial       "       -  26.88347,  " 

Hence,  the  logarithmic  mean  value  of  ho  is, 


log  hQ  =  -  25.13167,  ho  =  1.3542  X  10 


-25 


x-27 


Logarithm 

The  common  laboratory  value  is,   6.545    X  1(T"  —  27.81590 
The  solar  value  h  (earth)  X  T*'3 

(=  85.134),  5.5721  X  10"25  -  25.74602 

Computed  solar  value,  1.3542  X  10"25  -  25.13167 

We  collect  from  Bulletin  No.  4,  O.  M.  A.,  the  values  of  h 
as  computed  from  three  balloon  ascensions: 


—  I  —  -  —  I 
c     \      a      ) 


TABLE  39 
FROM  THREE  BALLOON  ASCENSIONS 


1911 

Uccle,  June  9 

Uccle,  Sept.  13 

Uccle,  Nov.  9 

Mean  Values 
up  to  60000 
Meters 

2  =  90000 

3.0212X10-20 
2.6632X10-25 
6.9295X10-26 
3.0141     ' 
1.7646     ' 
9.9955X10-27 
6.2650     ' 
5.3626     ' 
4.6475     ' 
3.2970     ' 

45.5100X10^7 
18.718 
10.2355 
6.3817 
5.4662 
4.5368 
3.2116 
6.545X10-27 

80000  
70000  ...  

3.1279X10-22 
1.3909X10-25 
6.0879X10-26 
1.9790     " 
1.0213     " 
6.3470X10-27 
5.4506     " 
4.4074     " 
3.1587     " 

5.3350X10-22 
2.5722X10-25* 
4.2185X10-26* 
1.0498     " 
6.5330X10-27 
5.5854     " 

60000 

50000  

40000  

30000  .... 

20000  

10000  
0000  

4.5545     " 
3.1791     " 
alue  of  h 

The  laboratory  v 

148  A   TREATISE   ON  THE   SUN'S   RADIATION 

The  value  of  h  from  balloon  ascensions  is  about  the  same  at 
30000  meters  as  the  laboratory  value.  The  terrestrial  value 
increases  upward,  as  does  the  corresponding  solar  value,  but 
more  rapidly.  If  the  change  from  terrestrial  values  of  h  is 
through  the  factor  J'*  =  (28.028)4/3  =  85.138  [1.93012],  to  the 
solar  value,  then  this  system  for  h  is  in  excellent  general  agree- 

r*  (      \ 
ment.     But  if  h  is  subject  to  changes  with  7  =  —  -,  —  -TT,  the 

ratio  of  the  gravitations,  it  cannot  be  regarded  as  a  universal 
constant.  It  is  apparently  a  complex  variable  having  significant 
values  in  radiation  processes  as  a  potential  coefficient. 

k  4/V  48  TT  a  \1/a 
In  the  formula  h  =  —   (  -  )    ,  c  is  the  velocity  of  light, 

3  X  1010  —  '-.  and  a  is  the  value  of  the  coefficient  in  u  =  a  T4 
sec.' 

derived  from  the  computed  KM  =  a-  Ta  in  (M.  K.  S.),  placing 
(<r,  a)  in  this  case  to  distinguish  it  from  the  other  notation. 

The  dimension  of  a  is  [  ~™~  *  T^  =  T~TZ  I  *-ne  kinetic  energy 

per  volume,  and  the  transformation  factor  from  (M.  K.  S.) 

1000 
to  (C.  G.  S.)  is  10  =  -r^.     We  now  identify  this  <r  (C.  G.  S.) 


with  a  of  the  usual  Stefan  formula,  /o  =  <r  T*,  since  both  repre- 
sent the  kinetic  energy  per  unit  volume.  It  expresses  the 
amount  of  radiant  energy  which  has  entered  the  volume  in 
one  second,  that  is,  the  amount  of  electromagnetic  energy  in 
a  column  whose  base  is  one  centimeter  square  and  length  is  c, 
the  velocity  of  light  per  second.  The  unit  volume,  if  spherical, 
has  four  times  the  surface  of  the  area  on  the  diameter.  Hence, 

a  =  —  ,  as  given  in   the   papers   on   radiation.       It  is   cus- 
c 

tomary  to  take  a  volume  energy  in  the  pure  ether  under  radiant 
agitation  a,  multiply  this  by  —  to  obtain  a  in  the  Stefan  formula. 

It  must  be  remembered  that  in  this  research  we  are  working 
in  the  opposite  direction,  from  the  effects  of  radiation  in  the 


COEFFICIENTS   IN   STEFAN  AND   WIEN-PLANCK  FORMULAS   149 


0 

u 

i 

X 

CD 

i 

00 

oo' 

00  OOOO 

o     m 

CO        N 

os'     os 

oo  os' 

oo 

05 

Tj<          OS          Tf«O 

TK        OS        TfCO 

co      co      cokn 
os     os'     osos 

05 

rHkO 
knrH 
NOO 

osos 

O    Ci  GO    OOOO                rH 
Oi   OOOO    00  W               Oi 

X 

0 

t- 

Tj* 
OO 

1  1 

rH         rH 

X     X 

:  :« 

V 

2                     3 

X           '  X 

X 

:  : 

kn  o 

&52 

co     o 

CD  CO 

O5 

o 

O5 

N-* 

1  "*. 

t- 

"*        N 

rHrH 

* 

00        00        t-t* 

kO 

TfCO 

r~ 

1 

I  . 

X 

X 

I 

X 

:       :       :  : 

x 

x" 

- 

OrH 

i  

| 

OS        t- 
CO        rH 

li 

rHrH 

os 

oo      kn      coco 

CO         CO         rHOS 
OS       OS       OSOO 

CO 

t- 

00 

co'ko 

N 

o 

CO 

1  ". 

00 

j 

i 

-< 

5 
J 

XX 

:       : 

:  : 

' 

X 

5. 

0 

X 

II 

X  X 

r)i 

i 
j 

H 
[4 

K) 

el 

i  1 

00 

i 

O       O        t-CO 

o     oo     co  kn 

1 

to  cc 

CiO^ 

rH    rH 
OS 

4 

coco 

r"1 

rH         OS         OSOS 

" 

c. 

i 

; 

4 

j 

«o 

X 

XX* 

5       : 

: 

i  =   =  = 

x 

= 

:  : 

x  x" 

t- 

kn  N 

i 

i 

3 

H 

1 

IIS 

oo      co 

CO        CD 

OSrH 

S8 

0 

i 

1C 

li 

05    rHrH    rH  CD 

N    OS 

1    oo' 

^ 

00 

i 

K 
K 

4 

^ 

J 

X 

i  ; 

X 

:   : 

: 

r       :       :  : 

' 

:  : 

"  x  x" 

CO 

§00 

3 

H 

1 

! 

Tf  t>CO 

sss 

oo     co 

t—       CD 

OS*       05 

CON 
OSOS 

05 

S-*          rHOO 
kO       UJrl* 

05        OS        OSOS 

kO 

•^ 

05 

NOS 

r)<CO 

OS  OS 

N     OSOO     t-rH         CO 

•<*  N  w  OON      t- 

N    Ot-    t-t-       0 

os  os'od  kn'kn     co 

T  od 

p| 

i 

1 

0 

a 

o 

H 
0 

j. 

ii  = 

XX 

010          Tj< 
T)<  TH            OS 

OTf      kn 

t-CD       0 

iis 

0048  ' 
9793  XIO-12 

li 

osos 

05 

t-        O       TjiOO 
OS        OS        0500 

CO 
CO 

g§ 
Z% 

S|: 

X  X  " 

os  oot-  cokn     N  •«*  coo  ~^- 
Tf  OON  coo      co  oo  t-  o 
N  eokn  cooo      N  t-  oso 
oo  t-co  us-.*     rn  oo  ON 

0    N 

m  co 

N    N 

1     OS 

i 

CON       rH 

rHrHrH 

rH         OS 

osos 

05 

OS       OS       OSOS 

O5 

osos 

OS    OSOS    OSOS       OS    Tjt    COrH 

c 
b 

— 

P 

j 

J 
J 

q 

< 

iii.     .          | 

xxx~  "     x 

TJI  CD  CO  O  t-        N 
S0050CO        CO 
OSOO  re  O       Tf 

11  TJI  os  t-eo     t- 

ooo 

°-CDCO 

o     kn 

t-        TJI 

II 

CO 

CO       O        t-CO 
N       00        COOS 

vo      kn      knkn 

0 

IO 

In 

§| 

~     2 

x  x" 

kn    NOS    COCO       N    t-    NCO    TfiOrH 

00     •*  OS     UOrH          rH     CO     Nil     mrHTjl 

TJI  Tj«eo  coco     o  kn  ooo  coknco 

Tjl     kO 

I  t-: 

f- 

ij 

•4 

NCOCON  rH         00. 

oo  oo  oo 

00       00 

oooo 

co 

OO       OO       00  00 

oo 

oooo 

00    OOOO    OOOO       OO    t-    OOOO    CDknrJi 

,5 

I,,,  I  : 

x"  '  "  x 

CONrHOTj-         O 

O5O  00  OO        CO 

o  oo  co  t-  o      co 
ooknrHCo     os 

rH  O5  CO 

os  oo  oo 

oo     oo 

li 

oooo 

I 

00          kn         COrH 

oo     oo     oooo 

g 

CO 

s§ 

oooo 

"  x  x"  " 

N    OO  kn    NOO       CO    CO    t-CO    H  OS  CO 

O  t-  kn  coo      os  N  NOS  ococo 
oo  t-t-  t-t-     in  TJI  NO  kn-^co 

§1 
§?§ 

1    oo 

CON  rH  rHOS         00 

oo  oo  oo 

00       OO 

oooo 

co 

00       00       00  00 

00 

oooo 

oo  oooo  oooo     oo  oo  oooo  cocokn 

NNrHrH 

N 

III 

§2 

0 

°\1 

i 

1 

i§ 
'7 

o  oo  oo     o  o  oo  ooo 

0    00    00        0    0    00    000 
|      ||rH        N^iCOOOONTji 

1       III   '-''-1'-1 

1  1  1  1 

^  

Above  Z,,  

150 


A  TREATISE   ON  THE   SUN  S  RADIATION 


kinetic  of  ponderable  gases  back  to  pure  ether  radiations.     We 
therefore  identify  our  computed  o-,  derived  from  atmospheric 

thermodynamic  data,  with  (o-)  immediately,  and  then  pass  to 

4. 
(a)  of  the  Stefan  formula  by  the  factor  —  ,  since  both  a-  and  a 


rML2     in 
must  have  the  same  dimensions  |_    T2    -  y~3     kinetic  energy  per 

unit  volume.     In  this  way  we  obtain  the  coefficients  Ci  =  8wc  h, 
€2  =  "T~,  in  the  Wien-Planck  formula  for  the  black  body  spec- 

trum.    Since  Ci  and  c2  are  both  variables  the  problem  becomes 
very  complex. 

We  have  for  diatomic  hydrogen  : 
Below  the  radiation  level  ...........  .........  8.8202  X  10~! 

Mean  value  in  the  isothermal  layers  ...........  9.1220  X  10~ 

TABLE  41 
COMPUTED  VALUES  OF  c\  =  8  v  c  h  FROM  THREE  BALLOON  ASCENSIONS 


1911 

Uccle,  June  9 

Uccle,  Sept.  13 

Uccle,  Nov.  9 

Mean  Values 
up  to  60000 
Meters 

3  =  90000 

2.2780X10-8 
2.0080X10-13 
5.2247X10'14 
2.2726      ' 
1.3304      ' 
7.5060X10-15 
4.7237      ' 
4.0435      ' 
3.5042      ' 
2.4859      ' 

26:7740X10-15 
14.1130 
7.7072 
4.8130 
4.1215 
3.4152 
2.4217 

80000 

2.3584X10-10 
1.0487X10-18 
3.0822X10-14 
1.4922     " 
7.7003X10-15 
4.7956     " 
4.1097     " 
3.3232     " 
2.3816     " 

70000  
60000  

4.0225X10-10* 
1.9394X10-13* 

50000  
40000  
30000  

3.1807X10-"* 
7.9153X10-15 
4.9258      " 
4.2112      " 
3.4182      " 
2.3976      " 

20000  

10000 

0000  

*  Omitted  in  the  means  on  account  of  the  unequal  heights. 

Ci  increases  with  the  height  in  both  of  the  atmospheres,  es- 
pecially in  the  upper  rarefied  gases. 

The  value  of  c\  computed  from  terrestrial  balloon  ascensions, 
at  the  top  of  the  isothermal  layer,  40000  meters,  agrees  at  that 
point  with  the  laboratory  value.  Reducing  this  to  the  sun  by 
the  gravitation  factor  7*  =  85.138  it  agrees  with  the  logarithmic 
mean  value,  quite  closely. 


COEFFICIENTS   IN   STEFAN  AND   WIEN-PLANCK  FORMULAS    151 


cot-  o  t-c* 
1-1  m  t-  rH  rH 
ooto  eo  t-co 


ooc- 
COrH 


oo     oo     oo  c- 


co  oo  in 

<O    U3  in 

o'  do 


CO  00  t-'*  CO 

O  03  t-K0  rH 

00  Tf  rH  t-  Tl< 

n  T-I  T-IO  o 


oow 


03    SS 

co  o'od 


<MOO 

rH  Oi 


§  §  s 


i-H  N03    CO 


OJ    5DO5  t-    Tjt 
i-H   O  T*  M   1-1 


5    •*    CO 

a  05  «o 

ss'g's 


00< 

<o  »n  < 

OrH< 


t-    M<0t£> 
rH    -rH  i-H  O 


IO    OO       r-t 


iH   »HO   O   O       O 

O    00    O    O        O 


<o  eo     o 

S3    S 


>O   •*    OOCO       "#    00( 

3  ^    "^     ^  in         t—     rH  C 
>O5    00    t-«O         rH     COC 


05CO  U5  OOi-H 
O3O5  Tj<  O5  IO 
THO  O  0501 


•*  00  CO  «5  O 
»-H  O3«O  00  rH 
t-  COO  COOO 


i-l    i-(OO    O5       O5    O5  O5    O5    O5       O5       O5  O5   O5 
05    050505    00       00    0000    00    00       00       0000   00 


0500    OOt-t- 


00  O5OCO  -^  IO  COCO  O  00  50  O5  CO  «D 

«O  TH  O3  rH  O5  OO  CO-^f  CO  O5  t-  -^  CO  O5 

t-  mo3rH  oo  t-  t- 1-  t-  (O  to  «o«o  in 

COeOOSfOCO  CO  CO"  CO  CO  CO  CO  COCOCOCOCO       COCOrHrHrH 

O5    O5OSO5    O5  O5  O5O5  O5  O5  O5  OS  O5    O5    O5  O5       O5    OS  O5    O5  O5 


N  ^O  rj<  00  «D  (O  rj<  O 
rH  U5  O  Tj(  OO  rH  IO  U3  00 
rjl  rHO5  <D  O3  rH  CO  O  lO 


O5    00    0000 


50C003 

OO  m  rH 

^8 


O       00    O 


COCOrHrH 

N 


OO 
OOO 

I  I    I    I 


i     i  i 


I     III 


152  A   TREATISE   ON  THE   SUN'S   RADIATION 

SUMMARY  OF  THE  RESULTS  FOR  ci=8  *•  ch. 


Number 

Logarithm 

General  mean  in  the  isothermal  layer  
(^1)2   Mean  just  above  the  radiation  layer 

9.1220X10-12 
8  4710  X  10"  12 

-12.96009 
—  12  92793 

(ci)o  Logarithmic  mean,  for  factor  0.461  
(ci)i    Mean  just  below  the  radiation  layer  
Balloon  ascensions  at  40000  meters  

8.2393X10-13 
5.7655X10-14 
7.7072X10"15 

-13.91589* 
-14.76084 
—  15  88690 

reduced  to  the  sun  .... 
Laboratory  value  reduced  to  the  sun  
Laboratory  value  at  the  earth 

6.5617X10-13 
4.2013X10-13 
4  9350  X10~15 

-13.81702* 
-13.62338* 
—  15  69329 

74/s  =  factor  for  c\  from  earth  to  sun  ..... 

85.138 

1.93012 

Compare  the  *  values. 

TABLE  43 

r  h 

COMPUTED  VALUES  OF  c2  =  ~r  FROM  THREE  BALLOON  ASCENSIONS 


1911 

Uccle, 
June  9 

Uccle, 
Sept.  13 

Uccle, 
Nov.  9 

Mean  Values 
up  to  60000 
Meters 

z  -  90000 

974  50 

80000 

252  67 

8  6066 

70000  

862  .  00* 

5  .  0559 

3.9188 

60000  

5.4771* 

2.6415 

2.4351 

2.5383 

50000 

1  9421* 

1  7583 

1  7106 

1  7345 

40000    .  .  . 

1  .  3820 

1.3694 

1.3591 

1  .  3702 

30000  

1  .  1555 

1  .  1432 

1  .  1401 

1  .  1496 

20000 

0  9716 

0  9674 

0  9668   » 

0  9686 

•     10000 

0  8310 

0.8318 

0.8299 

0.8309 

0000.  .  .  . 

0  .  7440 

0.7462 

0.7371 

0.7434 

cz  increases  with  the  height  in  both  of  the  atmospheres,  especially  in  the  upper  rarefied  gases. 

ch 
SUMMARY  OF  THE  RESULTS  FOR  cz  =  -r 


Number 

Logarithm 

General  mean  in  the  isothermal  layer  
(c2)2  Mean  just  above  the  radiation  layer  
(^2)0  Logarithmic  mean,  for  factor  0.461  
(^2)1  Mean  just  below  the  radiation  layer  
Balloon  ascensions  at  40000  meters  

96.117 
88.496 
40.787 
0.6296 
1.3702 

1.98280 
1.94692 
1.61052* 
-1.79906 
0.13678 

reduced  to  the  sun.  .      .  . 

4.1621 

0.61931* 

Laboratory  value  reduced  to  the  sun  
Laboratory  value  at  the  earth 

4.3834 
1.44303 

0.64181* 
0.15928 

T1/3  =  factor  for  C2,  from  earth  to  sun  

3.0376 

0.48253 

Compare  the  *  values. 


OT 

1 

COEFFICIENTS   IN  STEFAN  AND  WIEN-PLANCK  FORMULAS    15c 
X 

kO    Ot-H.    t-       CM    COOO    CM    kO       00       IO  CO    O   i-l  .         O    kO  O    OSOS         t- 
CM    lOOOO    t-       OO    COCO    O    t-       OS       .  .    CO    OOO         O    .  .    t-  CD         OS 
OS    OOOJO   »H       CO    t-  CM    CM    CM       O       CMOS    kO    kO  CM         t-    CM  i-H    COt-         CO 
00    OOOSiH    OO       OS    kOCO   .    t-       O       t-00    O    OSt-         CM    kOOO   OOOS         00 

1 

ll 

XX 

t-O 

coco 
Os'tM 

CMOS 

I'Sd 

tMCO 

o  «oko 
oo  CM' 

sll 

all 

0    || 

s  s§ 

c^Sl 
•^Jl  t-    • 

1  CM  : 

o  co  ^H 

1  0'_; 

t- 

cocot- 

CO       CO 

coco 

CO 

CO       t-       COCO 

t- 

coco      t-  t-t-  t-t-      co 

a 

I 

1 

|  2.4597X10-12 

:  ; 

1  a 

CM"     06 

IS 

00*00 

00 

X 

CO       O       OO 

t-     oo     OCM 

OO       O       1OCM 

00       O       i-I  CO* 

|s 

53 

I 

X 

00 

2 

X 

X 

CO       CM 
CM       kO 
CO       kO 

ko'kO 

§ 

kO 

£    S    S2 

co     TJ<     -*co 

OS       kO       OCD 

ko     co     t-  1-" 

CMtM         CM 

* 

.1  i. 

2.1100X1 
4.2333  ' 

CM       tl" 

CO  CM 

t-co 
oeo 
t-os 

00 

r-( 

I  S  S§ 

CO       CO       OSCM 

I 

kO 

X      X 

COCM         CO    S 

.          T)« 

*<« 

^ 

IO       kO       kOCO 

co 

COt-         <M    CM 

oo 

. 

i 

X 

A 

.    .           00.      .. 

X 

CO* 

X  X 

s 

:!: 

CM       OS 

li 

i 

CM       t-       i-HkO 
kO       CO       OOO 

10     us     ko'co 

CM  CO         kO    COOS    CO  kO 

co  eo       x  t-  os  CM  ^ 
eoko       CD  iftko  coco 

COCO         t-    OltM    CMCNJ 

u2 

I 

X 

t- 

Jo 

kO 

ll.   . 

ost-t- 

.'  .'  10 

•«*     t- 

00       CM 
kO       kO 

CMOS 

"CM 

i 

CO 

co'     co     co  co 

o 

i 

co 

X  X 

OS.         T-t   CMOS    t-.         IO 
CMOS         CM    kOOi    CM.         Tf 
..         00    COO    OOOS         »H 

COCO         COt-^OCCMCM         eO 

^ 

o. 

00.    . 

X 

t-00            I-H 

CMCO         IO 

cot-       eo 

CMOS         t- 

^H.              .' 

III 

II 

1 

t-     10     n«eo 
Tf      TJI      Tf  m 

g 
g 

X  X 

i-l  O         .    COCM    i-HO         kO    O    .O                h 
CMOS         COCMi-lOOS         kOOOt-t- 

ko  10      co  ooos  TH  eo       T-H  o  oo 

-t  kO  kO 

10       kO 

kOkO 

kO 

10       kO       kOkO 

in 

kOiO         10    lOkO    COCO         t-    CM   CMCM 

<fi 

XXX 

g^CMCOCO         kO 

eo  ko  co  co  co      oo 

sss 

eo     t- 

COkO  O   kO       O       CMCM 
OOTH    kO   00       CM       t-CM 
OSi-H   CM   CO       IO       COOO 

co  t-  t-  t-     t-     t-  1- 

s 
? 

.,    ,  .,  ..    III.  ,  ,  , 

XXX 

OO  00  00 

00       00 

0000   00  00       00       0000 

oo 

oooo      oo  osos  osos      I-H  CM  t-oo  ooosos 

i 

\ 

n. 

X 

OkOO 

loco  co 
t-  1-  1-' 

kO       00 
CD       CM 

t-     t- 
l>     t-' 

So  co 
t-t- 

CO 

s 

t- 
t-' 

00       O       t-kO 

n     os     kocM 

> 

X  X 

rH  OS         00   CO.    CMO         CO    .    COO    COOSCM 
CCM         CO    .CM    OOO         i-H    OS    CMCO    iH  rH  CO 

>00         00   OSOS    Oi-l         .    0    t^CM    OSOO 

t-       t-       t-t-   t-   t-t-         t-   t-t-   OOOO         00    OS    OSO    i-HCMCM 



ft!  ajaj 
N  NN 

1! 

N 

00  CO. 

§  s 

88 

o 

o     o     oo 

CM                  CM. 

1  1 

i         i    7   7  •*  f  °f  s^ 

154 


A   TREATISE   ON  THE   SUN  S  RADIATION 


d  Sit  40000  meters  reduced  to  the  sun  6.1657  X  10"13 
—  13.81702.  The  value  of  Ci  =  8  TT  c  h  increases  upward  from  the 
low  levels  to  the  vanishing  plane.  In  the  solar  atmosphere 
it  passes  through  a*  sudden  change  in  the  radiation  layer,  and 
its  mean  value,  derived  from  the  logarithmic  law  of  depletion, 
agrees  closely  with  the  value  of  c\  =  8  TT  c  h,  as  derived  from 
computations  of  the  terrestrial  thermodynamics,  at  the  height 
of  the  top  of  the  earth's  isothermal  layer. 


~15 


The  solar  value  (ci)0  8.2393  X  10~  is  the  same  as  in  the 
earth's  atmosphere  at  the  height  42000  meters  above  the  sea  level. 

The  value  of  cz,  computed  from  terrestrial  balloon  ascen- 
sions, agrees  at  the  top  of  the  isothermal  layer,  with  the 
laboratory  value  1.3702  and  1.4430,  respectively.  Reducing  the 
balloon  value  to  the  sun  by  the  factor  7^  =  3.0376,  it  becomes 
4.1621,  while  the  logarithmic  mean  value  by  the  factor  0.461 
is  40.787  at  the  sun,  about  ten  times  as  great.  The  same  abrupt 
change  occurs  in  c2  as  in  Ci  on  passing  the  radiation  layer,  0.62.96 
below  and  88.496  above  ZR  for  all  the  gases  computed.  At- 
tention is  called  to  diatomic  hydrogen,  which  agrees  with  all 
the  gases  above  this  layer,  but  is  very  different  below  it,  while 
monatomic  hydrogen  agrees  above  and  below  with  the  other 
gases.  It  is  supposed  that  this  indicates  dissociation  of  the 
hydrogen  diatomic  molecule  below  the  plane  of  radioactivity. 

TABLE  45 


COMPUTED  VALUES  OF  a  =  -       FROM  THREE  BALLOON  ASCENSIONS 


1911 

Uccle, 

Uccle, 

Uccle, 

Mean  Values 

June  9 

Sept.  13 

Nov.  9 

up  to  60000  Meters 

z-  90000... 

1  6403  X10~19 

80000 

2  2546  X10-17 

2  3765  X10~16 

70000. 

.7310X10-21* 

1.0422X10-15 

60000. 

.3995X10-15* 

4.1111     " 

4.1970     " 

4.1541X10-15 

7.  134  X  10-" 

50000. 

.4519X10-"* 

1.  0137X10-" 

1.  0091X10-" 

10.1140 

40000. 

:  .4091 

1.4221 

1.4287 

14.203 

g  3  rt  >-  Jj 

30000. 

.7945 

1.8195 

1.8153 

18.098 

|1||     | 

20000. 

3.0686 

3.0476 

3.0058 

30.407 

10000. 

4.6766 

4.5073 

4.7964 

47.268 

a)  >»«28  o""1 

0000. 

5.0810 

4.9880 

5  .  4694 

51  .  795 

HS.al-rfi 

*  Omitted  in  the  means. 

a  Decreases  with  the  height  in  the  atmospheres  of  the  earth  and  the  sun,  especially  in  the 
rarefied  gases  of  the  uppermost  strata. 


COEFFICIENTS  IN  STEFAN   AND  WIEN-PLANCK  FORMULAS    155 


i 

a>SS5  s  .; 

SooLcOCO        OS    N 

S     5     S         3         3    5     5 

00    rH    CO         00         TjfTjt    t- 
C-    lO    OO         OS         COrH    OS 
CO    rH    ^         rH         ^  t—    00 
OS    00    O         CO         OrH    CM 

TJ<  T)<  10      10      m  m  m 

lOCO         kO    OS  ^    kOCO         00 
CMCM         S    COCO    SS         CO 

in  m       in  in  m  m  m       in 

6.9846X10-* 
1.7970X10-2 
5.9479X10-8 
5.3028X10-9 

£8 

5 

o     ,    , 
X 

.     .     ,        .        ,00 
X  X 

:  : 

1000             • 

10  CD-* 

CM       CO    CO 

rH    in    CM        CM        Oi  CO    Tj< 
CO    OO    lO        CO        CM  -r*    Tjt 
rH    CM    CO         kO         COOS    00 

CD    CO    CO         t-        OOOS    rH 

Oi  O 
CD  00 

|     OSOO             • 

3 

0 

^ 

•«*s 

x 

X 

X 

og£ 

Tj<            t-     OS 

OO       00   CO 

co     i-i  os 

•*'       Tj»   CO* 

CO   CO    CM       «0       COOS    t- 
CM    CC    t-       O       00-<t    TJ< 

^  ^<  ^     ^      m  in  co 

SI   i 

|      rH  00 

* 

-d  d. 

8  10 

X 

X      X 

§S5 

r-tCO    CO       CO   CO 

CO    kO       CO       00  O    CM 
l>    OO    O       CM        -<S<CO   00 
CO    CO'   rjJ       tji       rjiTji   •«*' 

ir:  10        IH  TH 

'52 

XX*  " 

:    :    :      :      :  :    : 

XX*" 

SSI 

oo  in-<$  eo  co      oo  in 
o  t>co  t>  •»*     co  t- 

0    rH    CO         |0         00  rH    ^ 

CMCM         ^    OO    OSrJi 

t-o       to  co  10  cooo 
t-o»       t-  osos  osos 

kO  rH 

T 

.  .    .  .11.    . 

r-,N 

X 

X  X 

§11 

CO     OOrHrH     rH          CO    OS 
rH     CO  COCM     CO          Tj<     t- 

CO    CM    OS       kO       kOTf    CO 
rH    kO    00       CM       t-CM    tr- 
lO    kO    kO       CO       CO  t-    t- 

CMrH           CO    Tj«  Tf    OOO           OS 
0000           rH    IOO    rHCM           CO 

COtM 

ii      

dd.. 

£^ 

XX 

os  co       eo  coco'  eo  co"     ^'  rj< 

X  X 

rHrH           CM    CO^    C0t2     .      CO    kO    SkO 

O  i-l  O 
le^g 

1     ko'rn' 

000  

dd 

CM 

XXX 

OSlOOOSrH          OS    OOO    O       kO    CO 

co  o  t-  oo  10       oo  OrHCM  co     co  eo 

••tfOSCOkOCM         O    rHCMCO    •*        kO    kO 

t>   00    OS       O       CMU5   00 

»  eo  co     ^<     ko  co  t> 
b  ko  10     ko     ko  10  10 

xx 

I-IT)<       oo  kOtM  osco       t-  TI«  OSCM  eoeoN 

OCM         t-    OOO   rHCM         kO    CO    ^  OS    CMt-CM 

lii| 

CM  CO  CM  -^  lO         CO    COCOCO    CO       CO    CO 

to  co  co     to     coco  co 

«oco      «o  «o«o  t-t-      t-  os  koco  eococo 

c 
o 

d  

dd.. 

'ZJ  2 

.2  v 

^ 

X 

O    OS    CM       CO        rH  t~    CM 

rH    rH    CM         CM         CO  CO    ** 
00    00   00       00        OO  00    OO 

X  X 

CMCO         t-   lOCO    rHOi         i-(   <O    t-O    t-OSOS 

t-cM       t-  ooos  oo       <o  rji  coko  eoooTf 

Tl<  kO         t-    CM  t-   COOO         ^Jt    CO    OO    CftOCM 
0000         00   OSOS   OO         CO    t-    COt-    Tl<kOlO 

....  -9000 
7.7050 
1.4937 
m  below  the  rac 
ic  isothermal  la> 

ifi 

OOOOO         O    OOO    O        O   O 

N 

77  T 

1     I             1             1           rH         CM    •<*    COOO    OCM^ 
1              1              1              1              1        1        1      1     rHrHrH 

1   1  1  1 

:  •§£ 

156 


A  TREATISE   ON  THE   SUN'S   RADIATION 


SUMMARY  OF  THE  RESULTS  FOR  a  =  — — l 


Number 

Logarithm 

General  mean  in  the  isothermal  layer 

6  8397  XlO-19 

—  19  83504 

02  Mean  just  #bove  the  radiation  layer  . 

9  3  127  X10~19 

—  19  96907 

a0  Logarithmic  mean,  for  the  factor  0.461  
a,\  Mean  just  below  the  radiation  layer  

7.  5738  XlO-15 
2  3960  XlO-12 

-15.87932* 
—  12  37949 

Balloon  ascensions  at  55000  meters 

7  390  XlO-15 

—  15  86864* 

reduced  to  the  sun  
Laboratory  value  reduced  to  the  sun  

7.  390  XlO-15 
7  3900  XlO-15 

-15.86864 
—  15  86864 

Laboratory  value  at  the  earth 

7  3900  XlO-15 

_ie;  86864* 

Compare  the  *  values. 

The  y  factor  is   1.00,  since 


gives  v13  /  y'3  =  1.00, 


so  that  for  the  coefficient  a  the  reduction  from  the  values  at  the 
earth  to  those  corresponding  with  them  at  the  sun  disappears. 
The  value  of  a  changes  with  the  height  in  each  atmosphere. 

TABLE  47 

COMPUTED  VALUES  OF  a  =  ^  FROM  THREE  BALLOON  ASCENSIONS 

4 


1911 

Uccle, 
June  9 

Uccle, 
Sept.  13 

Uccle, 
Nov.  9 

Mean  Values 
up  to  60000  Meters 

2  =  90000.... 
80000.... 

1.2302X10-9 
1.7824X10-6 
1.0790X10-5 
3.1478     " 
7.5883     " 
1.0716  XlO-4 
1.3613     " 
2.2543     " 
3.5973     " 
4.1020     " 

3.1156X10-5 
7.5957     " 
1.0650  XlO-1 
1.3572     " 
2.2805     " 
3.4934     " 
3.5510     " 

5.  3557  XIO-8 

1    0)  -U   OJ   1) 

2  =>  ««  fcjS 

PlM 

2.8183X10-* 
7.  81  63  XlO-6 
3.0834X10-5 
7.6030     " 
1.  0666X10-* 
1.3646     " 
2.2856     " 
3.3805     " 
3.7410     " 

70000.    . 
60000.    . 
50000.    . 
40000.    . 
30000.   . 
20000.    . 
10000.    . 
0000... 

3  .  5482  XlG-u* 
1.0496X10-5* 
1.0889  XlO-4* 
1.0568 
1.3458     ' 
2.3015     ' 
3.5075     ' 
3.8101     ' 

*  Omitted  from  the  means. 

<r  Decreases  upwards  in  the  atmospheres  of  the  sun  and  the  earth. 

The  laboratory  value  7.390  X  10  ~15  is  encountered  again  at 
about  55000  meters  above  the  sea  level;  it  is,  also,  found  in  the 
radiation  layers  near  the  bottom  of  the  isothermal  layer  of 
each  solar  gas.  It  seems  as  if  the  solar  radiation  once  originated 
advances  through  the  solar  envelope  without  much  absorption, 
being  depleted  chiefly  by  scattering,  and  that  it  arrives  un- 
changed at  the  outer  layers  of  the  earth's  gaseous  envelope. 


COEFFICIENTS    IN    STEFAN    AND   WIEN-PLANCK    FORMULAS 


157 


• 


x" 

ff\-*4t  tf*     — /     PC-J     *-*>  tO     \r\     rtS    kA    CO  h—     CO    CA-vK     •*#     \ft  ~-4     -«-H  O 

II 


t-  OOrJtCO  TH  CO  O  lO 

oo  Oicoeo  O  o  TH  CM 

t-  COCM^  to  CO  t- CM 

CO  COtOCO  CO  Tj«  OrH 


X 

10  •»*  OCM  CD 

Oi    Oi  Oi  CO  O 

Oi  oi  oooo  o 

CM    00  00  oo'  oi 


X 

O    t-    Tj<  C-    O 


t-  O  coco 


)  00    Oi    Oi 
1 10    to   00 


Jl 


X 

O    00 


CM    t-    CM  CO    00 


TH    CO 

Tf      O 


CO    t-    t-00    00 


£ 

X 

S3°,THg£8 

Oi    *tl«    CO  00    Oi    TH  CO 
t-    O   CO  CO    Oi    CO  CO 

kO   CO    COCO    CO    t~t> 


X  X 

00   CMCMTH 

SS?c$S 

TH  OOCMCO 


CO  W 
OOO 


CO   CO  CO   CO    CO  t-   OO 


>Tf      OiOi 

IS  S3 


t-  Tj<  OOf-l 

OO  O  (M  IO 

O  t-  CMt- 

iH  CO  •**  ^J 

CO  CO  CO  CO 


t-t- 

coco 


OiOi    CO    Oit- 


COCO   b-   t-00 


x 

OTH 


11 


rH  Oi  iOrH  t-eo 

t-  t-  Oir-l     CM^ 

t-  CO  Tf  CO    i-lOi 

iOi  O  CM^*   COt- 


TH  to    to    10  IO  IO    to   tO    tOtO    IO    IO    IO    IO  lO    IO    to  IO   CO   COCO   COCO    t- 


111. 
xxx" 


CO  10  CO  CO  t- 


O  to  CO  COO 
CO  CO  CD  CO  t> 


THOOO  CO  O  TjfOi  ^  Oi  r}<  tot 
THCOO  to  00  CMCO  TH  IO  O  COC 
COt-CM  CD  CO  IOCO  00  Oi  TH  CM  • 


THCO  CM   OiC 

lOrH  CM    COI 

t-  Oi  t-  eo 

TfTjl  kO     TH 


X 

t-Oi 


Oi    O    00 

rH    O    TH 


00   OO  00  Oi   Oi   Oi 


Oi   OiO>   Oi   OiOi   Oi   OiOi   THiH   TH    TH    00  Oi 


"a 


O    OrH  CM_    CM    CO    COCO    CO 


II 

CO  CO 

oooo 


TI«TH  oo  oeo  tot- 

OOOi  CM    O  t-  ^TH 

coco  -^  to  to  cot- 

0000  00    OOOO  OOOO 


XX 

O   CO    tOCO 

oo  to  t-  T}< 

§t-    ^"O 
CO   OTH 


If  III  §  HI  ||  is  5  %  °  %%  s  i||  ||  ||  I  § 

lOOtOOtO   TH  ^  ''I'|||||THCMT}< 

CMCMi-lTH  III 

N  « 


0134X 
2546X 
5270X 
6329X 


s  ooo 
I    ooto 


Sco  to 
TfO 

CM  OOO 
CM  t-TH 
I  *  * 


;  i 
1  g 

Jl 


2' 


Above  Z  »  ...... 
Below.  .7  ...... 
Diatomic  hydrog 
Mean  value  in  th 


158 


A   TREATISE   ON  THE   SUN'S   RADIATION 


The  excessive  changes  in  a  above  55000  meters  involves 
problems  of  difficulty  in  the  kinetic  theory  of  gases.  From  the 
level  55000  meter,  where  a  =  7.390  X  10  ~15,  to  the  sea  level, 
the  value  of  a  steadily  increases  to  51.795  X  10  ~15,  where  it  is 
about  seven  times  as  large  as  at  55000  meters.  This  is,  however, 
compensated  by  a  change  of  the  exponent  from  a  =  4.00  to 
a  =  3.82  on  the  sea  level. 


SUMMARY  OF  THE  RESULTS  FOR  <r  =  -^ 


Number 

Logarithm 

General  mean  in  the  isothermal  layer    . 

5  3028  X10~9 

—9  72450 

<72  Mean  just  above  the  radiation  layer  
<7o  Logarithmic  mean,  for  the  factor  0.461  
o~i  Mean  just  below  the  radiation  layer  

6.9846X10"9 
5.6425X10"5 
1.7970X10~2 

-9.84414 
-5.75147* 
-2.25455 

Balloon  ascensions  at  55000  meters  
reduced  to  the  sun  

5.3557XHT5 
5  3557  XHT5 

-5.72882* 
-5  72882 

Laboratory  value  reduced  to  the  sun  
Laboratory  value  at  the  earth 

5.5424XKT6 
5  5424  X10~5 

-5.74370* 
—5  74370 

Compare  the  *  values. 

The  7  factor  =  1.00  for  a  and  for  #,  since  these  have 
the  same  dimensions.  These  values  of  a-  should  be  compared 
with  those  in  Table  26,  from  which  it  is  seen  that  there  is  close 
agreement  in  the  results.  The  ordinary  laboratory  value 
5.5424  X  10~5  is  found  again  near  the  level  55000  meters  in 
the  earth's  atmosphere,  and,  also,  in  the  radiation  layer  of  the 
solar  isothermal  strata  of  the  several  gases.  The  exceptional 
behavior  of  diatomic  hydrogen  H2  =  2.00  below  the  radiation 
layer  should  be  noted. 

It  should  be  remembered  that  we  have  taken  a  and  a  to 
represent  a  certain  amount  of  kinetic  energy  per  volume,  accord- 


[  __ 

T2     '      TZ 
J~  J-*t 


-i 


The  first  form  is 


ing  to  the  dimensions    |      T*  '  L?  ~ LT*\' 

more  instructive  arid  it  should  always  be  employed  in  the  dis- 
cussion of  the  radiation  problems.     Since  work  =  force  times 


length, 


ML 


MD 


j  the  amount  of  work  done  is  equivalent 


COEFFICIENTS   IN   STEFAN  AND   WIEN-PLANCK  FORMULAS    159 


to  the  kinetic  energy  per  volume.     The  practical  units  of  work 
are, 

kilogram  meter2 
second2        ' 


(M.  K.  S.)  Joule  = 


(C.  G.  S.)  Erg      = 


gram  centimeter2 
second2         .' 


The  dimensions  of  a  and  a  both  give     T2  .    —     the  work 

JL  JL/ 

Joule,    erg.  Joule        ,    erg 

per  volume,  * — ; — ,  — r.     The  interpretation  •*-     -  and  - 

vol.      vol.  area  area 

is  erroneous  from  every  point  of  view. 


60o- 


TABLE  49 

COMPUTED  VALUES  FROM  THREE  BALLOON  ASCENSIONS,  :1^-,  IN  GRAM 
CALORIES  PER  SQUARE  CENTIMETER  PER  MINUTE 


1911 

Uccle, 
June  9 

Uccle, 
Sept.  13 

Uccle, 
Nov.  9 

Means 

3  90000. 

1.7637X10-15 

80000. 

2.3250X10-13 

2.5553X10-12 



70000. 

5.0869X10-17 

1.1206X10-H 

1.5469X10-11 

— 

60000. 

1.5047X10-" 

4.4204     " 

4.5127     " 

3.4793X10-11 

7.9667X10-11 

50000. 

1.5611X10-io 

1.0900X10-1" 

1.0850X10-io 

1.2454X10-io 

40000. 

1.5151 

1  .  5291 

1  .  5720 

1  .  5387 

ii^si 

30000. 

1  .  9294 

1.9564 

1.9518 

1  .  9459 

llss 

20000. 

2.2995 

3.2768 

3.2319 

3.2694 

S     l|«   . 

10000. 

.  . 

5.0285 

4.8453 

5.1573 

5.0104 

^b°l|l 

0000. 

5.4633 

5.3630 

5.8809 

5.3691 

H3.28-3! 

The  corresponding  flux  per  unit  area  is  given  instead  of  the  volume  kinetic  energy. 

60  o-      gram  calories 


SUMMARY  OF  THE  RESULTS  FOR 


cm.2  minute 


Number 

Logarithm 

General  mean  in  the  isothermal  layer. 
(60  crA)2  Mean  just  above  the  radiation  layer.  . 
(60  O"A)O  Logarithmic  mean,  for  the  factor  0.461 
(60  <TA)\  Mean  just  below  the  radiation  layer.  . 
Balloon  ascensions  at  55000  meters  .  . 
reduced  to  the  sun 
Laboratory  value  reduced  to  the  sun  . 
Laboratory  value  at  the  earth  

7.6329X10-15 
10.  0134  X  10-  15 
7.6015X10-11 
2.2546X10-8 
7.9667X10-11 
7.9667X10-11 
7.900X10-11 
7.900X10-11 

-15.88269 
-14.00058 
-11.88090* 
-  8.35307 
-11.90127* 
-11.90127 
-11.89763 
-11.89763* 

Compare  the  *  values. 


160 


A  TREATISE   ON  THE   SUN'S   RADIATION 


As  in  the  case  of  a,  <r,  we  have  —j-  ,  in  gram  calories  per 

square  centimeter  per  minute,  having  about  the  same  values  in 
the  laboratory,  at  55000  meters  in  the  atmosphere,  and  at  the 
radiation  layer  in  the  solar  gases. 

Check  on  the  Computations 

We  have  now  computed  —  r—  in  two  different  ways,  quite 

A 

distinct  from  one  another: 

(1)  The  value  of  c  from  pairs  (log  c,  a)  in  the  (M.  K.  S.) 

system  was  reduced  directly  to  —  :—  by  the  factor  7 

=  0.000014336  [-  5.15644]. 

(2)  The  data  of  the  general  law  P  =  p  R  T  were  reduced 
from  the  (M.  K.  S.)  to  the  (C.  G.  S.)  system;  from  these  were 


computed  K  .  N  .  k  .  h  .  ci,  c2,  and  finally  a 
60  a-  c        1 


* 


r^i 


TABLE   50 

EXAMPLES  OF  THE  AGREEMENT  OF  60^  AS  COMPUTED  BY  Two  DIFFERENT 

METHODS 


UCCLE,  SEPT.  13,  1911 

HELIUM  ON  THE  SUN 

CADMIUM  ON  THE  SUN 

z 

z 

z 
Meters 

Direct 

Indirect 

Kilo- 
met. 

Direct 

Indirect 

Kilo- 
met. 

Direct 

Indirect 

80000.. 

-13.36644 

-13.36644 

10000 

-15.12234 

-15.12235 

300 

-15.52855 

-15.52854 

70000. 

-11.04944 

-11.04945 

5000 

-15.72819 

-15.72819 

200 

-15.82711 

-15.82711 

60000  . 

-11.64544 

-11.64546 

1000 

-15.70684 

-15.70683 

100 

-15.77835 

-15.77835 

50000. 

-10.03744 

-10.03742 

500 

-15.73344 

-15.73345 

40 

-15.77304 

-15.77305 

40000. 

-10.18444 

-10.18443 

0 

-15.76984 

-15.76984 

0 

-15.84725 

-15.84724 

30000. 

-10.29144 

-10.29145 

-500 

-15.79714 

-15.79715 

-40 

-15.91579 

-15.91580 

20000. 

-10.51544 

-10.51545 

-1500 

-15.83214 

-15.83215 

-100 

-8.43548 

-8.43547 

10000. 

-10.68544 

-10.68542 

-3000 

-8.33440 

-8.33442 

-140 

-8.44294 

-8.44296 

0000. 

-10.72944 

-10.72943 

-5000 

-8.34084 

-8.34082 

-200 

-8.44737 

-8.44736 

The  check  is  complete  in  the  logarithms,  and  indeed  this 
forms  a  convenient  check  upon  the  long  intervening  numerical 


COEFFICIENTS  IN  STEFAN  AND  WIEN-PLANCK  FORMULAS    161 

processes.     It  fully  disproves  Very's  contention  that  the  reduc- 
tion factor  should  be, 

1000  X  60         0>0014336  [_  3.15644]. 


4.1851  X  107 

His  factor  violates  the  law  of  dimensions,  and  leads  to  im- 
possible conditions,  incompatible  with  solar  and  terrestrial 
atmospheric  physics.  He  has  transferred  the  factor  of  the 
surface  integral  in  the  Poynting  equation  to  the  volume  in- 
tegral, which  is  erroneous. 

Second   Computation   of  the    Wien-Planck    Coefficients,    Taking 
N  =  constant  (6.062  X  1023)  and  EQ  =  variable. 

The  preceding  Tables,  36  to  50,  have  been  computed  on 
the  assumption  that  the  kinetic  energy  of  one  molecule  is  a 
constant,  E0  =  4.3769  X  10~n,  so  that  N  and  the  other  quan- 
tities dependent  upon  it  become  variables.  The  difficulty 
raised  by  this  dilemma  has  been  stated  on  page  140.  We  have 
in  Tables  51  and  53  compiled  the  data  by  using  N  =  constant 
(6.062  X  1023)  in  the  formulas  for  k,  h,  c\,  c2,  E0,  n,  instead  of 
the  EQ  =  constant,  as  in  Table  34.  Table  52  is  for  the  solar 
elements,  and  Table  53  is  for  atmospheric  air  in  three  balloon 
ascensions.  The  heights  z  for  each  element  are  in  part  the 
same  as  those  in  the  Tables  36-50,  and  by  comparing  the  re- 
spective tables  it  can  easily  be  seen  what  has  been  the  effect 

of  making  this  change.     The  values  of  a  =  — —  are  identical 

£2  ' 

in  both  computations,  so  that  the  Stefan  Law  JQ  =  a  T*  re- 
mains unaltered  in  its  values.  The  distribution  in  the  spectrum 
becomes  somewhat  different,  due  to  the  changes  in  c\  and  c2. 
It  is  remarkable  that  while  c\  is  larger  below  the  plane  of  the 
source  of  the  solar  radiation  zs,  and  smaller  above  it,  they  nearly 
agree  in  value  on  that  plane.  That  is  to  say,  the  N  =  constant 
system  is  about  equal  to  the  E0  =  constant  system  for  the 
critical  strata  in  the  solar  atmosphere  where  the  black  radiation 
is  generated.  Similarly,  c2  is  larger  below  and  smaller  above 


162 


A   TREATISE   ON   THE    SUN'S   RADIATION 


the  planes  ZR,  but  nearly  equal  on  that  plane  for  the  N  =  con- 
stant and  the  E0  =  constant  systems.  This  is  seen  in  Table  51, 
where  the  final  values  are  as  follows: 


STT  c  h,  N  =  constant -  13.83601 

EQ  =  constant.  ,    —  13.84623 


k 


N  =  constant 
EQ  =  constant 


Means 
Laboratory  value 


Means -  13.84112 

Laboratory  value ^. . .   <  -  13.62338 

.  ch 

log  c2  =  -r, 


0.89025 
0.87570 


0.88298 
0.64181 


TABLE  51 

COMPARISON  OF  LOG  ci,  LOG  c2,  AT  THE  RADIATION  LEVELS,  AS  COMPUTED 

WITH    N  =  CONSTANT    (HEAVY    TYPE)    AND    WITH    E0  =  CONSTANT 

(LIGHT  TYPE) 


LOG  Ci  =  8*  ch 

Means 

ch 

LOG  cz  =  -r 

K 

Element 

Below 

Above 

Below 

Above 

Means 

Hi  =1.00  

-14.81319 

-12.83545 

-13.82432 

-1.83162 

1  .  90906 

0.87034 

-14.81319 

-12.90829 

-13.86074 

-1.83162 

1  .  92727 

0.87945 

#2=2.00  

-13.99743 

-12.82635 

-12.41189* 

0.97706 

1.88327 

1.43017* 

-13.94548 

-12.87856 

-12.41202* 

0.96467 

1.89632 

1.43050* 

H  =4  

-14.88971 

-12.88834 

-13.88903 

-1.84879 

1  .  96140 

0.90510 

-14.68828 

-12.96015 

-13.82422 

-1.79843 

1.97935 

0.88889 

C   =12....... 

-14.76247 

-12.85613 

-13.80930 

-1.78090 

1  .  94005 

0.86047 

-14.76247 

-12.94243 

-13.85245 

-1.78090 

1.96162 

0.87126 

Ca  =40.  

-14.87619 

-12.72973 

-13.80296 

-1.82750 

1  .  95076 

0.88913 

-14.70853 

-12.95001 

-13.82927 

-1.77764 

1  .  96922 

0.87343 

Zw=65  

-14.89982 

—  12.84379 

-13   87181 

—  1.82960 

1.95123 

0.89042 

-14.78310 

-12.96068 

-13.87189 

-1.80042 

1.98045 

0  .  89044 

Cd=112  

-14.78527 

-12.86050 

-13.82289 

-1.80370 

1  .  93985 

0.87178 

-14.78527 

-12.94107 

-13.86317 

-1.80370 

1.95999 

0.88185 

Hg=198 

—  14.84221 

—12.80124 

-13   82173 

—1.81595 

2.07311 

0.94453 

-14.77151 

-12.87220 

-13.82186 

-1.79828 

1  .  89085 

0.84457 

General  mean  for  N  =constant.  .  .    —  13  .  83601 

General  mean  for  2V  =  constant  0  .  89025 

"  Eo=constant.  .    —13.84623 

"   Eo=constant  0.87570 

Average  for  both                                —13  841  1  2 

Average  for  hnth                        0  88298 

Laboratory  vali 

ie 

1  Q     £QQQQ 

.  .  .     —  lo  .  bZooo 

Laboratory 

value 

0.64181 

COEFFICIENTS  IN  STEFAN  AND  WIEN-PLANCK  FORMULAS    163 


The  general  result  is  that  the  Wien-Planck  formula, 
STJ-EA        Sirch  1 


ch 


X5 


has  reproduced  the  usual  laboratory  values  of  Ci,  c2,  approxi- 
mately, at  the  critical  places  of  the  generation  of  the  black 
radiation  in  the  solar  atmospheres  of  the  monatomic  gases. 
For  the  diatomic  hydrogen  the  values  of  log  c\  and  log  Cz  are 

TABLE  52 

SECOND  COMPUTATION  OF  THE  WIEN-PLANCK  COEFFICIENTS,  TAKING  N= 

CONSTANT  (6.062  XlO23)  AND  EQ= VARIABLE 

Hydrogen  (Monatomic,  .Hi  =  1.00) 


2 

k 

h 

c\ 

C2 

Eo 

n 

26000  .  .  . 

4.9256X10-16 

8.6208X10-25 

6.5000X10-13 

52.506 

2.  3272  XlO-12 

1.1933X1016 

24000  .  .  . 

5.9027     " 

1.0819  XlO-24 

8.1578     " 

54.990 

3.0988 

1  .  8335 

20000  .  .  . 

7.8382     " 

1.5520     " 

1.1701  XlO-12 

59.399 

5.0554 

3  .  8207 

16000... 

9.8098     " 

2.0810     " 

1.5691     " 

63  .  641 

7.5780 

7.0117 

14000... 

1.0769  XlO-15 

2.  3579  i    ' 

1.7778     " 

65  .  687 

9  .  0456 

9.1445 

10000... 

1.2489     " 

2.8762      ' 

2.1686     " 

69.087 

1.2364  XlO-11 

1.4613X1017 

6000... 

1.4490     " 

3.4737      ' 

2.6191     " 

71  .  920 

1.6192 

2.1900 

4000... 

1.6005     " 

3.8934      ' 

2.9355     " 

72.980 

1  .  8365 

2.6454 

2000... 

1.8028     " 

4.4488      ' 

3.3543     " 

74.028 

2.0796 

3.1874 

0... 

2.0368     " 

5.1160      ' 

3.8574     " 

75.355 

2.3528 

3.8356 

-2000.  .  . 

2.3108      ' 

5.8831     " 

4.4358     " 

76.383 

2  .  6656 

4.6255 

-4000.  .  . 

2.6155      ' 

6.7940     " 

5.1226     " 

77  .  930 

3  .  0134 

5  .  5597 

-6000.  .  . 

2.9671      ' 

7.8345     " 

5.9070     " 

79.214 

3.4141 

6.7046 

-8000.  .. 

3.3588      ' 

9.0800X10-24 

6.  8462  XlO-12 

81  .  108 

3.8596 

8.0586 

-10000.  .. 

3.8135      ' 

8.6264X10-26 

6.  5134X10-" 

0.67861 

4.3769 

9.7324 

-12000... 

4.2966      ' 

1.0079X10-25 

7.5990     " 

0  .  70370 

4.9562 

1.1727X1018 

-14000.  .  . 

4.3882      ' 

1.0331     " 

7.7890     " 

0.70623 

5.5789 

1.4006     " 

Hydrogen  (Diatomic,  #2 =2.00) 


z 

k 

h 

c\ 

C2 

E0 

n 

25000. 

.5898  XlO-1' 

1.0563X10-27 

7.  9645X10-" 

199.33 

1.9073  XlO-17 

2.1450X102 

20000. 

.6208  XlO-1* 

4.2837X10-25 

3.  2292X10-" 

79.287 

2.3221X10-" 

2.4403Xioi2 

15000. 

.5557     " 

1.8554X10-24 

1.3989X10-12 

73.670 

2.3291X10-12 

7.  1175X10" 

10000. 

.5397X10-1* 

3.8151     " 

2.8764 

74.333 

8.0910     " 

1.4794XlO'6 

5000. 

.9279     " 

4.9243     " 

3.7129 

76.627 

1.7033X10-" 

9.5363     " 

1000. 

.2183     " 

5.6586     " 

4.2665 

76.527 

2.5555 

2.5879X1017 

0. 

2.4254     " 

6.2094     " 

4.6818 

76.772 

2.7915 

3  .  1220 

-1000. 

2.6588     " 

6.8247     " 

5.1456 

77.003 

3.0549 

4.0161 

-2000. 

2.9075     " 

7.4135     " 

5.5896 

76.493 

3.3414 

5.0075 

-4000. 

3.4901     " 

8.8918     " 

6.  7043  XlO-12 

76.432 

3.9997 

6.1927 

-6000  . 

4.1642     " 

1.3185     " 

9.9410X10-1' 

9.4986 

4.7884 

1.1859 

-8000. 

4  .  7642     " 

1  .  5241     " 

1.1491  XlO-12 

9.5973 

5.7169 

1.8783 

-10000. 

4  .  8734     " 

1.5609     " 

1.1769     " 

9.6086 

6.7155 

2.7928 

-12000. 

4  .  8734     " 

1.5573     " 

1  .  1742     " 

9  .  5865 

7.7197 

3.9342 

-14000. 

4  .  8734     " 

1.5454     " 

1.1652     " 

9  .  5130 

8.7238 

5.3160 

164 


A  TREATISE   ON  THE   SUN?S  RADIATION 

TABLE  52—  Continued 

Helium,  w=4 


z 

k 

h 

Cl 

C2 

So 

M 

10000... 

2.3627X10-17 

2.4814X10-26 

1.8710X10-" 

31  .  509 

2.6225X10-" 

1.4275X1013 

5000... 

8.3012XlO-i« 

1.7937X10-2* 

1.3524X10-12 

64.923 

4.7070X10-12 

3.4323X10" 

2000  .  .  . 

1.4058X10-16 

3.7343     " 

2.8156     " 

79.692 

1.4234X10-11 

1.8050X1017 

1000  .  .  . 

1.6192     " 

4.4438     " 

3.4286     " 

82.332 

1  .  8459     " 

2.6656     " 

0... 

2.0444     " 

5.7778     " 

4.3563     " 

84.786 

2.3628     " 

3.8603     " 

-1000... 

2.6210     " 

7.6713     " 

5.7840     " 

87.806 

3.0215     " 

5.5820     " 

-2000.  .  . 

3.3628     " 

1.0256X10-23 

7.7328     " 

91.496 

3.8665     " 

8.0802     " 

-4000  .  .  . 

4.3720     " 

1.0288X10-25 

7.7573     " 

0.70598 

6.1979     " 

1  .  6400X1018 

-6000.  .. 

4.3720     " 

1.0256     " 

7.7327     " 

0.70373 

8.6926     " 

2.7241     " 

-7000.  .  . 

4.3720     " 

1.0135     " 

7.7172     " 

0.70233 

9.9405     " 

3.3310     " 

Carbon,  m  =  12 


z 

k 

ft 

Cl 

C2 

Eo 

n 

1000  

1.1494X10-15 

2.8513X10-2* 

2.1499X10-12 

74.418 

1.1121X10-11 

1.2464X1017 

800.... 

1.2552     " 

3.1986     " 

2.4116     " 

76.448 

1.3179     " 

1  .  6080     " 

600.... 

1.3713     " 

3.5568     " 

2.6817     " 

77.650 

1  .  5427     " 

2.0364     " 

400.... 

1  .  5490     " 

4.0875     " 

3.0819     " 

79.164 

1  .  7925     " 

2.5506     " 

200.... 

1.8016     " 

4.8216     " 

3.6354     " 

80.288 

2.0822     " 

3  .  1932     " 

0.... 

2.0899     " 

5.7181     " 

4.3113     " 

82.080 

2.4122     " 

3.9819     " 

-200.... 

2.4308     " 

6.7633     " 

5.0994     " 

83.470 

2.8021     " 

4.9850     " 

-400.... 

2.8198     " 

8.0402     " 

6.0621     " 

85.540 

3.2463     " 

6.2164     " 

-600  

3.2798     " 

9.5230X10-24 

7.1802X10-12 

87.106 

3.7708     " 

7.7825     " 

-800  

3.8135     " 

7.6756X10-26 

5.7873X10-14 

0.60381 

4.3769     " 

9.7324X" 

-1000.  ... 

4.4354     " 

9.2620     " 

6.9833     " 

0.62646 

5.0829     " 

1.2179X1018 

-2000.... 

5.2888     " 

1.1456X10-25 

8.6378     " 

0.64984 

9.4995     " 

2.9039     " 

Calcium,  w=40 


z 

k 

h 

Cl 

C2 

Eo 

n 

1000  

5.7870X10-1" 

4.6618X10-27 

3.5148X10-15 

24  .  167 

2.6041X10-15 

4.4671X1011 

800.... 

1.9029XlO-i« 

2.7974X10-25 

2.1092X10-13 

44.102 

4.2815X10-13 

9.4174X10" 

600.... 

5.7680     " 

1.1172X10-24 

8.4232     " 

57.924 

2.5602X10-12 

1.3771X10" 

400.  ... 

1.0112X10-15 

2.3329     " 

1.7589X10-12 

69.212 

7.1287     " 

6.3981     " 

200.... 

1.3173     " 

3.4023     " 

2.5652     " 

77.482 

1.3436X10-" 

1  .  6553X10" 

100.... 

1  .  5573     " 

4.1731     " 

3.1464     " 

80.393 

1  .  7519     " 

2.4644     " 

0  

1.9462     " 

5.3485     "• 

4.0326     " 

82.446 

2.2493     " 

3  .  5853     " 

-100  

2.4990     " 

7.1182     " 

5.3670X10-12 

85.452 

2.8809     " 

5.1966     " 

-200.... 

3.3512     " 

.9730X10-24 

7.5195X10-W 

89.283 

3  .  8531     " 

8.0497     " 

-400  

4.8558     " 

.0880X10-25 

8.2035     " 

0.67220 

6.2630     " 

1  .  6658X1018 

-600.... 

4.8558     " 

.0843     " 

8.1754     " 

0.66990 

9.0452     " 

2.8913     " 

-800.... 

4.8558     " 

.0807     " 

8.1480     " 

0.66765 

1.1822XlO-io 

4.3203     " 

-1000  

4.8558     " 

.0780    ." 

8.1280     " 

0.66601 

1.4580     " 

5.9273     " 

-2000  

4  .  8558     " 

1  .  0742     " 

8.0992     " 

0.66365 

2.8459     " 

1.6134X10" 

COEFFICIENTS  IN  STEFAN  AND  WIEN-PLANCK  FORMULAS    165 


TABLE  52 — Continued 
Zinc,  w=65 


z 

k 

h 

c\ 

C1 

Eo 

n 

600..      . 

4.1704X10-17 

4.4253X10-26 

3.3365X10-14 

31  .  833 

4.0037X10-M 

2.6926X10" 

400.  .      . 

5.2404XlO-i« 

1.0250X10-2" 

7.7280X10-13 

58.677 

2.4523X10-12 

1.2909XlQi« 

200.  .      . 

1.1361X10-15 

2.8737     " 

2.1667X10-12 

75.882 

1.0122X10-" 

1.0825X1017 

100..      . 

1.4277     " 

3.8275     " 

2.8859     " 

80.427 

1.6061     " 

2.1636     " 

0..      . 

2.0693     " 

5.8203     " 

4.3883     " 

84.378 

2.3947     " 

3.9386     " 

-100..    .. 

3.1069     " 

9.2562X10-24 

6.9790X10-12 

89.378 

3.5768     " 

7.1897     " 

-200..    .. 

4.6771     " 

1.0531X10-25 

7.9400X10-14 

0.67546 

5.3544     " 

1.3169X1018 

Cadmium,  m  =  112 


z 

k 

h 

Cl 

c-i 

Eo 

n 

200.  .      . 

7.4130X10-16 

1.4297X10-24 

1.0780X10-12 

57.859 

3.2243X10-12 

1.9470XlQi« 

100..      . 

1.3961X10-15 

3.4522     " 

2.6029     " 

74.177 

7.9580     " 

1.3862X1Q17 

80.  . 

1.4332     " 

3.6483     " 

2.7507     " 

76.366 

.4188X10-11 

1.7963     " 

60.  . 

1  .  5095     " 

3.9150     " 

2.9518     " 

77.803 

.1020     " 

2.2536     " 

40.. 

1.6601     " 

4.3665     " 

3.2922     " 

78.906 

.2700     " 

2.7946     «• 

20.. 

1.8970     " 

5.0824     " 

3.8320     " 

80.375 

.4588     " 

3.4403     " 

0.. 

2.1827     " 

5.9411     " 

4.4795     " 

81  .  658 

.6772     " 

4.2413     " 

-20.. 

2.5116     " 

6.9680     " 

5.2537     " 

83.420 

1  .  9237     " 

5.2101     " 

-40.  . 

2  .  8843     " 

8.1738     " 

6.1629     " 

85.016 

2.2119     " 

6.4231     " 

-60.  . 

3.3145     " 

9.6192X10-24 

7.2527X10-12 

87.066 

2.5981     " 

7.7193     " 

-80.  . 

3.8135     " 

8.0893X10-26 

6.0991X10-14 

0.63636 

4.3769     " 

9.7324     " 

-100.  . 

4.3397     " 

9.4570     " 

7.1303     " 

0.65375 

5.0254     " 

1.1974X1018 

-200.  . 

4.4926     " 

9.8138     " 

7.3995     " 

0.65533 

8.5872     " 

2.6744     " 

Mercury,  w  =  198 


z 

k 

h 

Cl 

cz 

So 

n 

200  ..      . 

2.6278X10-17 

2.1953X10-26 

1.6552X10-14 

25.062 

1.8161X10-15 

2.6013X10" 

100..      . 

8.1208X10-16 

1.4672X10-24 

1.1063X10-12 

54.203 

4.8116X10-12 

3.5475X1Q18 

80.  .      . 

9.9204     " 

1.9254     " 

1.-4517     " 

58.225 

7.3663     " 

6.7196     " 

60..      . 

1.1787X10-15 

2.4217     " 

1.8259     " 

61.637 

1.0431X10-11 

1.1323X1017 

40.  .      . 

1.3882     " 

2.9922     " 

2.2561     " 

64  .  662 

1.4056 

1.7711     " 

20.  .      . 

1.6529     " 

3.7052     " 

2.7936     " 

67.248 

1  .  8347 

2.6473     " 

0.  .      . 

2.0469     " 

4.7212     " 

3  .  5597     " 

69.196 

2.3549 

3.8409     " 

-20..      . 

2.6258     " 

6.2974     " 

4.7481     " 

71.948 

3.0181 

5.5729     " 

-40..      . 

3.3720     " 

8.3922X10-24 

6.3276X10-12 

118.33 

3.8721 

8.0980     " 

-60..      . 

4.2269     " 

9.2225X10-26 

6.9537X10-14 

0.65456 

4.9454 

1.1688X1018 

-80..      . 

4.3282     " 

9.4758     " 

7.1438     " 

0.65680 

6.1636 

1.6263     " 

-100.  .      . 

4.3282     " 

9.4488     " 

7.1243     " 

0.65493 

7.3885 

2.1345     " 

quite  different  from  the  means  just  computed.  A  study  of 
Tables  52  and  53,  above  and  below  the  radiation  level  ZR,  and 
an  intercomparison  of  the  entire  N  =  constant  system  with  the 
EQ  =  constant  system,  make  it  difficult  to  decide  whether  one 
is  better  than  the  other,  since  the  divergence  is  about  the  .same 
in  opposite  directions. 


166 


A   TREATISE  ON  THE   SUN  S  RADIATION 
TABLE  53 


SECOND  COMPUTATION  OF  THE  WIEN- PLANCK  COEFFICIENTS  TAKING 
N  =  CONSTANT  (6.062  XlO23)  AND  E0= VARIABLE 

Balloon  Ascension,  Uccle,  June  9,  1911 


z 

k 

h 

Cl 

C2 

Eo 

n 

70000.    ... 

2.2849XlO-a> 

7.0356X10-30 

5.3049X10-18 

9.23760 

2.4145X10-21 

1.6448X10 

60000.    ... 

6.  8192X10-  18 

2.  1060  XlO-28 

1.5879X10-16 

0.92648 

2.6598X10-16 

4.2395X10" 

50000.    ... 

3.  6806  XlO-17 

9.1420     " 

6.8928     " 

0.74515 

3  .  1470X10-14 

1.4747X1015 

40000.    ... 

4.1715     " 

1.0911  XlO-27 

8.2268     " 

0.78470 

1.0199     " 

3.3560X1017 

30000  .    ... 

4.9893     " 

1.2780     " 

9.6358     " 

0.76844 

1.6389     " 

1.  0788  X  10" 

20000.    ... 

7.9440     " 

1.9870     " 

1.4982X10-15 

0.75040 

2.5665     " 

3.2545     " 

10000.    ... 

1.1963  XlO-16 

2  .  9803     " 

2.2471     " 

0.74738 

4.0538     " 

1.0027X1019 

000.    ... 

1.3598     " 

3.4419     " 

2.5955     " 

0.75890 

3.9417     " 

2.5393     " 

Balloon  Ascension,  Uccle,  September  13,  1911 


z 

k 

h 

Cl 

cz 

Eo 

n 

80000  
70000  
60000      .  .  . 

1.9762X10-19 
5.  4412  XlO-18 
1.4529  XlO-17 
2.9574     " 
4.2809     " 
5.0749     " 
8.0032     " 
1.1778X10-16 
1.3606     " 

2.9057X10-29 
1.7195X10-28 
4.0310     " 
7.6977     " 
1.1260X10-25 
1.3013     " 
2.0114     " 
2.9542     " 
3.4632     " 

2.  1909  XlO-17 
1.2965X10-16 
3  .  0393     " 
5  .  8039     " 
8.4898     " 
9.8118     " 
1.5165X10-" 
2.2274     " 
2.6111     " 

4.41110 
0.94805 
0.83240 
0.78085 
0.82626 
0.76927 
0.75397 
0.75265 
0.76357 

2.9648X1Q-19 
3.6735X10-16 
1.7435  XlO-15 
4.8800     " 
1.0661  X10~14 
1  .  6976     " 
2.6380     " 
4.1275     " 
5.9699     " 

2.2120X106 
9.0932X1013 
4.2047X1015 
5.2979X1016 
3.  6261  XlO17 
1.1399     " 
3.3742X1018 
1.0157X1019 
2.5191     " 

50000 

40000  

30000  

20000  
10000  
000  

Balloon  Ascension,  Uccle,  November  9,  1911 


z 

k 

h 

Cl 

C2 

Eo 

n 

90000  

7.0927X10-20 

9.7702X10-30 

7.  3665  XlO-18 

4.1325 

4.2489X10-19 

6.4684X106 

80000  

1.7830X10-18 

6.3581X10-2* 

4.7939X10-17 

1  .  0698 

1.0701X10-16 

5.2278X1012 

70000  

6.1459     V 

1.8166X10-28 

1.3696X10-16 

0.88672 

6.4536     " 

4.3616X1014 

60000  

1.3574  XlO-17 

3  .  6568     " 

2.7571     " 

0.80816 

2.  0368  XlO-15 

7.3798X1015 

50000  

2  .  7384     " 

6.9577     " 

5.2459     " 

0.76223 

4.9301     " 

7.2954X1016 

40000  

3.9570     " 

1.0123  XlO-27 

7.6325     " 

0  .  76747 

1.  0033X10-" 

3.  7383  XlO17 

30000  

4.7053     " 

1  .  1774     " 

8.8774     " 

0.75609 

1  .  5904     " 

1.1619X1018 

20000  

7.3330     " 

1  .  7982     " 

1.3559X10-15 

0.73568 

2.4552     " 

3.3843     " 

10000  

1.1528X10-16 

2.8129     " 

2.1209     " 

0.73202 

3.8242     " 

1.0098X1019 

000  

1.3606     " 

3.3585     " 

2.5322     " 

0.74048 

5.6494     " 

2.6326     " 

Table  52  summarizes  the  following  results  for  the  N  — 
constant  system:  The  Boltzmann  entropy  coefficient  dimin- 
ishes with  the  height,  without  any  per  saltum  change  at  the 
level  of  radiation  ZR;  the  same  is  true  of  the  kinetic  energy 
of  one  molecule  E0,  and  for  the  number  of  molecules  per  unit 
volume  n.  On  the  other  hand,  the  Planck  Wirkungsquantum  h, 


COEFFICIENTS  IN  STEFAN  AND  WIEN-PLANCK  FORMULAS     167 

and  the  Wien-Planck  coefficients  all  change  abruptly  at  the 
radiation  levels  ZR.  In  making  up  the  mean  change  for  any 
of  the  chemical  elements  it  is  sufficient  to  add  to  the  lower 
term  0.461  (upper  .  .  .  lower)  instead  of  taking  the  direct 
arithmetical  mean.  Table  53,  for  the  earth's  atmosphere,  shows 
that  there  is  a  gradual  diminution  in  the  values  of  h,  c\,  c2,  as 
well  as  in  those  for  k,  Eo,  n.  The  intercomparison  between  the 
solar  and  the  terrestrial  elements  can  be  readily  studied,  and 
applied  in  any  theoretical  discussions  regarding  the  laws  that 
govern  the  processes  of  radiation. 

The  foregoing  data  have  been  computed  directly  from  the 
non-adiabatic  thermodynamics  of  the  atmospheres  of  the  sun 
and  the  earth,  and  these  represent  the  accomplished  effects  of 
radiation  upon  the  gases  per  unit  volume,  as  summarized  in  the 
temperature  which  is  the  general  parameter.  We  shall  now 
proceed  to  give  the  data  relating  to  the  electromagnetic  energy 
more  directly,  in  order  to  be  able  to  examine  the  distribution 
of  the  radiant  energy — that  is,  its  concentration,  or  its  depar- 
ture from  the  normal  value  in  the  aether,  outside  the  atmos- 
pheric gaseous  media.  It  has  been  explained  that  the  nearly 
perfect  solar  black  radiation  seems  to  arrive  at  the  levels  about 
50000  meters  above  the  sea  level  without  much  change  in  its 
conditions,  and  that  there  is  a  progressive  concentration  from 
that  height  down  to  the  sea  level. 

NOTE. — The  coefficient  h  is  to  be  regarded  as  indicating  the  potential  energy 
controlling  the  movements  in  h  v,  where  v  is  the  frequency,  while  k  r  is  a 
general  constant  in  a  given  atmosphere,  and  represents  the  corresponding 
kinetic  energy  of  a  single  atom  or  electron. 


CHAPTER  V 

The  Elements  of  Black  Radiation  in  the  Atmospheres  of  the 
Sun  and  the  Earth 

The  Concentrations  of  Black  Radiation  in  Gaseous  Media 

WE  must  distinguish  clearly  between  the  three  stages  of 
the  physical  conversion  of  the  black  body  radiation  as  electro- 
magnetic energy  in  the  aether,  outside  the  gaseous  media,  and 
the  final  effect  as  temperature  of  the  gaseous  media  through 
the  kinetic  energy  of  the  molecules.  In  any  given  volume  of 
the  gaseous  media,  (1)  there  is  absorbed  a  certain  amount  of 
radiant  energy,  and  this  is  not  a  variable  amount  so  long  as  the 
temperature  is  steady;  (2)  this  constant  amount  of  energy  per 

fM  L?     1  ~j 
— ^ —  YS  \  *s  maintained  by  a  flux  of  energy  entering 

and  leaving  the  volume  in  a  constant  flow  per  unit  time,  with 

rM .  L2    i    L-\     rMn  f  * 

the  velocity  of  light,  [  ^2  •  j-z  •  y  I  =  [^  J ,  from  the  non- 
gaseous  volumes;  (3)  there  is  a  marked  difference  in  the  volume 
density  of  the  constant  aether-flux  through  it,  and  of  the  volume 
kinetic  energy  at  different  levels  in  the  atmosphere  as  developed 
by  the  computations,  having  the  nature  of  concentration,  or 
convergence  and  divergence  of  the  energy,  away  from  the  simple 
black  radiation  of  the  aether  itself.  These  changes  in  concen- 
tration are  expressions  of  the  operation  of  many  physical  proc- 
esses, measured  in  the  bulk,  and  although  these  processes  are 
not  understood,  it  may  be  of  importance  to  give  the  correspond- 
ing quantities  which  come  immediately  from  the  general  for- 
mulas of  radiation,  as  summarized  by  Planck  in  Die  Theorie  der 
Warmestrahlung,  Leipzig,  1913.  The  following  computations 
depend  entirely  upon  the  volume  density,  u  =  a  T*,  where  T 
is  the  temperature  of  the  volume,  and  the  volume  energy  per 

168 


ELEMENTS   OF  BLACK  RADIATION  IN  THE  ATMOSPHERES    169 

deg.4  is  taken  from  the  thermodynamic  computations,  as  in  Table 
22.  The  coefficient  (c)  of  that  table  is  identical  with  a-  per  unit 

volume;   and  a  =  — ,  this  c  being  velocity  .of  light.     This  trans- 
c 

ition  from  my  thermodynamic  notation  to  the  Planck 
notation  should  cause  no  confusion.  The  solar  radiation  on 
entering  the  earth's  atmosphere  cannot  be  treated  in  the 
simple  manner  just  indicated,  because  all  of  the  electromagnetic 
radiant  energy  is  not  converted  into  molecular  kinetic  energy 
per  unit  volume,  as  determined  by  the  steady  temperature 
that  is  observed.  Rather,  the  incoming  radiation  separates 
into  three  branches,  (l)  the  absorbed  part  making  temperature; 
(2)  the  scattered  or  reflected  part  that  does  not  make  tem- 
perature; (3)  the  free  radiation  of  high-frequency  that  pene- 
trates to  the  sea  level,  and  is  there  converted  into  a  lower  fre- 
quency, to  return  with  a  further  complex  absorption  and  scatter- 
ing to  space.  These  processes  have  been  indicated  in  the  Treatise, 
also  in  Bulletin  No.  4,  0.  M.  A.  The  reader  should  bear  the 
line  of  our  argument  in  mind.  Having  computed  KW  =  c  Ta 
from  the  thermodynamic  temperature  conditions  of  the  atmo- 
sphere, the  temperature  representing  the  final  absorptions  of 
all  kinds,  where  (c)  differs  from  the  black  body  o-,  and  (a)  is 

different  from  4.00,  we  next  passed  from  (a)  to  4.00,  retaining 

4 

the  same  value  of  (c),  as  a  variable  <r,  and  computed  u  =  —  .  a  T4, 

c 

with  the  other  dependent  quantities.  Our  hypothesis  consists  in 
supposing  that  by  changing  the  exponent  from  its  local  volume 
value  (a)  to  the  black  body  value  4.00,  we  thereby  have  added 
to  the  absorbed  radiant  energy,  first,  the  part  which  was  scattered 
before  reaching  the  sea  level,  and  secondly,  the  remaining  free 
radiation  of  the  heat  current.  Our  procedure  is  justified  by  the 
fact  that  the  thermodynamic  data,  at  the  earth  and  the  sun, 
agree  with  the  simplest  interpretation  of  the  bolometer  ordi- 
nates,  and  with  the  interpretation  of  the  pyrheliometric  data, 
which  closely  identify  it  with  the  theory  of  scattering  that  is 
associated  with  the  number  of  molecules  per  unit  volume,  or 
the  local* density .  The  difficulties  inherent  in  these  problems 


170 

make  it  necessary  to  take  certain  tentative  steps  in  computation, 
which  must  be  tested  by  comparison  with  the  facts  of  ob- 
servation. If  the  volume  aether  energy,  a  =  —  a,  of  black  solar 

radiation  were  to  be  utilized  completely  in  the  gaseous  media 
to  make  temperature,  without  concentration  into  or  leakage 
from  the  perfect  mirror  enclosure  containing  the  given  volume, 
this  process  of  proceeding  from  temperature  T,  and  KWj  first 

to  <TJ  and  then  to  a  =  — ,  would  produce  the  primitive  aether 

c 

volume  energy  for  each  unit  volume  in  the  column,  which  is 
3  X  1010  centimeters  in  length.  As  a  matter  of  fact,  the  (a)  thus 
computed  in  u  =  a  T4,  does  vary  from  the  laboratory  standard 
value,  and  this  must  signify  that  the  radiant  energy  has  con- 
centration into  or  leakage  from  the  gaseous  volume,  which  does 
not  act  as  if  it  were  perfectly  enclosed.  The  following  tables 
exhibit  the  results  of  such  computations.  If  the  exact  con- 
ditions that  they  represent  in  the  atmospheres  of  the  sun  and  the 
earth  are  understood,  they  are  valuable  as  representing  the 
departures  or  concentrations  in  atmospheres  in  reference  to  the 
standard  values  of  the  aether  outside  of  gaseous  atmospheres. 
It  may  be  possible  from  them  to  make  some  further  advances 
toward  the  fundamental  laws  of  radiation  within  ordinary 
gaseous  media,  whether  on  the  basis  of  dynamics,  or  electro- 
magnetics, or  the  electrons  in  atoms  and  molecules. 

The  Poynting  equation  summarizes  these  leakage  processes: 


At  ZR,  T  =  7654.°5,  log  T4  =  15.53568,  log  u  =  1.40063, 
u  =  2.5155  X  10. 

It  is  difficult  to  extrapolate  the  values  of  u,  as  computed  at 
the  assigned  heights,  to  the  corresponding  values  at  zRj  in 
order  to  compute  the  mean  u  (ZR). 

The  ratio  factor  is  obtained  directly  by  dividing  the  value 
of  u  above  ZR  by  the  value  of  u  below  ZR. 


ELEMENTS   OF  BLACK  RADIATION  IN  THE  ATMOSPHERES       171 


£2 


XX        X 

O       COt-i-lt-t 

CO        CO  •«*  "*  OS  C 

os     eocoincoc 

T}«       OOSOS  t-  C 


XXX 

co     co  **"  os  eg  •* 


0-* 
0-3 


33 


X     X 

•  trioooco     co 


t-     eg»oiooo»o     in 

5!    S£3g8§    8 


•^     ioo5.-ir-icg 


XX        X 

•^  -rf  C^  t>  O       CO 

oi  si  c^i  oo  oi     <D 


222    2= 

XXX     X 


-S  S2 

x  xx 

)O^*  ^O 

i  co  eg  oo  oo 


2. 

xxxx* 


0-3 

0* 

05 


x"  "  "  " 
js^^^ 

•^<  CON  i-t 

WD  CO  C^  00 


XX     X 

ci  i-J  CD  ,-J  •*' 


ll; 

xx" 


IO  t-  t-  t-t- 

NOO  rfi  O  O 


OOrHlOOOC 

t-'  T-l  r-I  r4  eg     eg  eg  eg'  eg  eg     eg 


XXX 

i-HNcgcgoq     coSt-So 
eg  eg  N  o>  «-J 


63X10-8 
54X10-* 


X 

t-oo  t-coi 

OS  OS  OS  OS 


XX* 


OS  TfOS  rl<OS 
TjtOlOT-HO 
CO^f  r)<  U5U5 
OS  OS  OS  O5  OS 


sSS 


lim 

xxxxx 

oco  »-icgo 
cg^foost- 


is 

X 
•^c, . 

0"*l 
»-llO(_ 

ooooooos< 


X 

10,-it-coos     t-oocoooosoo 

t^eg^cg?     S^coooo"^1 
Oi-ii-Hcgcg     CD  « t- co  co  eg  t- 


^Tj-ioosoo     cgegcgegcg     eocococoeo     co     cococoeoco     cocoeococo     co  -^  eg  eo  co  "-1  oo 


22-  S- 
xx"  x" 


«0  eg  10  i-l  eg 


XX    X 

co  osiHco»ot-  oocgcoo^  CD  10 co ao oo  t- eg 
eg  rj<t-os»-!co  "*  t-  os  eg  •**  OOSCOOOCDWOS 
co  eo  co  co -*f  •*!<  lot-oscg^  CD  co  t-  n  co  co  TT 


cgegcgegcg     eg  co  eo  w  CD  t-  rn 


S§§   88 


Sdo'dd  o'  o' 
o  o  o  o  o  o 
o  o  o  o  o  oo 
eg  •«*  CD  oo  o  eg  ^ 
1    1    1    I 


172 


A   TREATISE   ON  THE   SUN?S  RADIATION 


§   w 


'  ti 


«c  «o  co  co  oo  o  «-H 

rH   0   t-   0   00   Tf   0 

r-i  co  t-  •*  T-I  eo  o 
<o  i-!  Tji  csi  T)*  <o  t^ 


11.1.1 

^-ir-i:    ^i:    ^H:    :    : 
XX      X      X 


St^c^ooob-^t-i-iio 

§  O  00  C$  00  CO  CO  £?  00 
10  l>  t-'  00  00  OS  0>  05  O> 


eoeoTi' 


'    '    '  Tf  •*  t-  to  m  n  co 

o  oo  esi  •*  as  to  o 

Tt  oo  o]  co  10  co  oo 

05    t-  »H    Tf  0    N    Tf 

N   rH  r-i   •*  t>   T-i   CO 


XXXXXX 


«>  CO  •<*  C<    »H         t-  »-"  CO 


xxxxx 


.  , 

1 


xxxxx    x 


ELEMENTS  OF   BLACK  RADIATION  IN  THE  ATMOSPHERES    173 

The  balloon  volume  density  is  about  4.63  times  that  of 
the  laboratory  (a).  The  former  concentrates  by  this  factor,  but 
it  is  compensated  by  lowering  the  exponent  from  4.00  to  3.80. 

The  computations  for  u  =  a  T*  =  — .  a  T*  are  as  follows: 

From  Table  6  take  the  temperature  T  of  the  element  at 
the  height  indicated,  and  compute  log  T4;  from  Table  22  take 
log  c  in  the  (M.  K.  S.)  system,  and  convert  it  into  log  a  in  the 

(C.  G.  S.)  system  by  the  factor  10;  multiply  a  by  -|-  =  1.3333  X 

10-10,    [log  -j-  -  10.12494],  to  produce  (a);  finally,  compute  the 

volume  density  u.  There  are  several  remarks  regarding  the 
values  of  u  in  the  solar  atmosphere,  Table  54,  and  in  the  ter- 
restrial atmosphere,  Table  55.  Above  the  transition  level  ZR 
the  exponent  in  u  =  a  T4  resulted  at  about  the  Stefan  black 
body  value,  4.00,  as  in  Tables  18-22,  so  that  the  low-efficiency 
values  of  Table  54  are  those  which  actually  exist.  The  ob- 
served thermal  efficiency  of  the  inner  corona  during  eclipses 
has  been  singularly  low.  Below  ZR  the  exponent  averages 
about  2.420,  Table  24,  so  that  the  actual  efficiency  passes 
through  the  radiation  levels  with  a  simple  gradient.  The  po- 
tential efficiency  is,  however,  very  great  below  ZR,  as  is  seen 
from  Table  54  by  using  the  exponent  4.00.  The  imprisoned 
potential  radiation  changes  abruptly  by  a  factor  whose  mean 
value  is, 

Factor  of  radiation  =  3.2442  X  10~7  (monatomic) 
"  "          =  1.5880  X  10~3  (diatomic) 

This  is  not  quite  correct,  because  the  values  of  u  should 
be  extrapolated  to  the  exact  heights  ZR,  and  this  it  is  difficult 
to  do  practically  without  additional  computations. 

The  mean  value  of  u  on  the  photosphere  is  up  =  2.4512  X 

io-3. 

In  the  earth's  atmosphere  the  first  section  of  Table  55  gives 
a  few  values  computed  from  balloon  ascensions  and  the  mean 
values  up  to  60000  meters;  in  the  second  section  the  values 
are  computed  from  the  mean  temperatures  Tt  and  log  a  = 


174  A   TREATISE   ON  THE   SUN'S   RADIATION 

-  15.86494  from  Table  26.  It  is  noted  that  the  balloon  and  the 
laboratory  values  coincide  at  about  50000  meters,  but  that  the 
former  concentrate  sevenfold  at  the  sea  level. 

The  incoming  solar  radiation,  therefore,  undergoes  a  process 
of  concentration  in  the  successive  volumes  from  50000  meters 
to  the  sea  level,  until  it  becomes  4.63  times  as  dense  per  unit 
volume  as  the  density  computed  directly  from  the  laboratory 
value,  from  log  a  =  —  15.86494  and  the  exponent  4.00.  This 
result  should  not  be  misinterpreted.  For  with  the  concen- 
tration of  the  coefficient  there  is  a  diminution  of  the  exponent, 

4(T 

from  4.00  to  3.80,  so  that  the  effective  ua  =  —  T3'80    remains 

c 

about  the  same  as  if  there  had  been  no  concentration  of  the 
coefficient.     Take  an  example  from  the  isothermal  region,  for 
T  =  220°  and  the  exponent  3.72,  computing  for  both  cases: 
Black  radiation,  u  =  [-  15.86497]  X  2204-00  = 

1.716  X  10  ~5; 

Concentrated  coefficient)  i5.86497]  X  4.63  X  220^  = 

with  smaller  exponent,  J 

1.754  X  10  ~5. 

A  very  slight  adjustment  would  equalize  those  results.  Similar 
conditions  prevail  on  the  sun.  Take  the  data  from.  Table  24, 
for  T  =  7655°. 

Nearly  black  radiation,    u  =  [-  9.81409]  X  -  X  76554-053  = 

c 

4.795  X  10  -*; 

Concentrated  coefficientl    ^  =  {_  22mQ]  x  4 
with  smaller  exponent,  J  c 

5.362  X  10  ~3. 

Here,  again,  small  adjustments  would  produce  an  equality. 
This  concentration  at  the  sun  is  [6.42610],  or  conversely,  the  dim- 
inution of  the  coefficient  is  [-  7.57390]  =  3.749  X  10~7,  while 
that  given  on  Table  54,  which  is  not  quite  accurate,  is  3.2442  X 
10~7.  This  remarkable  interplay  between  coefficient  and  ex- 
ponent we  have  interpreted  as  connected  with  radiation  in  the 
solar  atmosphere  at  the  levels  ZR,  with  further  scattering  in  the 
upper  levels,  amounting  to  an  equivalent  1.87  gr.  cal./cm.2  min. 


ELEMENTS   OF   BLACK  RADIATION  IN   THE  ATMOSPHERES      175 


•a 


I  "'' 


°°,  .  ° 

X  XX~  "  X 

00  OCOrHOlO  CO 

at  Tt  rHooosio  co 

O  1O^J<  rHOS  t-  CO 

CO  t-COOO  t-00  rH 

00  COrHCOWOS  rH 


XX    X 


rHrHCQCOrH 


X    X 

«O       COOSt-OCM 


iH       rHCO-^COCO       t- 


ig,  5 

XX"  X 

o  oo  o  t— in  10 

rH  00  *  ^  O  N 

oo'  od  o»  N"  eo'  oi 


X10-8 
X10-5 
X10-4 


OJCOCO       CSIOOOCO-^       CJ 
CO  CO  t-        t-  rH  Tt i  00  N       CO 

•<j<rHrH        Tj<  IO  IO  U5  CO        Co' 


--i  SS 

x  xx 

ilflCOfc*.  IOO 

>OO  CD  f  NCO 

>t>0>rH  ON 


cot>t>t>ao      NCO 


isiLi 

xxx"  x 

CD  OS  CD  t- ^  t- O5  i-H  CO  IO 

OOC4OiOi  •<!<  TJ<  IO  lO  lO 

OOrHCXJOOl  rHTj«t-OCO 

rHCDrHOOOO  »O  U5  1C  CO  CO 


8422X10-* 
6889X103 
0613X10* 
0763  " 
4226X105 


«0t-t-t>t>        OO^CNCO! 


rHcoeoeoN     lOrHiocnco     t> 

COOSOt-T}<         rHlOOprHlO        00 

cocoooco 


rHCOW5t»Oi  O- 

rHCO  IO  t-  OJ  rH  • 

ooioogos  co  eo< 

OJCOT}«TjtU5  OS  I 


XX 


00       X 


XX 

iHOO 


3    g{ 


ill 

XXX 


rHCO        U5U5kOlOCD 


Jiiil 
xxxxx 


X  X 

O»  OJ  OS  OS  OS         rH  rH*  , 


rHOO^fCOt-         kONOOTfO         t- 00  CM  O  OS  CO  t- 

sssss  sssss  s?s§3?a 


i=i=: 
X     X 

O5  lO  (M  O  t- 
CN  OS  t-OSO 

CM  CO  rHCO  t- 


CO  Tj<  O  CO  N         ^COt—^rH        00 
IO  CM  OS  IO  CJ        lO  CM  00  IO  CM       00 

t-WOSOSO       OOOOrH       rH 


X    XX 


00  O  rH  PH  N  CO  ** 


0000000004 


i  i  i 


§§§§§  g: 

CMTtCOOOO       O< 

I     I     I      I  r-t  N  •<}<  CO  00  O  CM  rt 

I      I      I      '      I  I      I      I      I    rHrHrH 


I  I      I      I      I 


XX 

^*  00 
^J*  00 
d  iO 


176 


A  TREATISE   ON  THE   SUN  S  RADIATION 


at  the  distance  of  the  earth.  Similarly,  in  the  earth's  atmos- 
phere, we  retained  the  coefficient  unchanged,  but  by  making 
the  exponent  4.00  in  place  of  that  found  in  the  pair  value  (log  c.  a) 
we  passed  from  the  selective  absorption  Ja  to  the  black  /0, 
which  added  the  amount  scattered  and  the  free  heat,  since 
Ja  =  1.46  and  J0  =  3.95  calories. 

The  pressure  of  the  radiation  energy  per  unit  volume  is 
taken  p  =  f  u,  so  that  all  the  conditions  that  were  described 
under  u  can  be  transferred  to  p.  The  factor  of  transition  at 
ZR  is  the  same;  p  (photosphere)  =  8.1707  X  10"4. 

TABLE  57 
VALUES  OF  p  IN  THE  EARTH'S  ATMOSPHERE 


1911 

June  9 

Sept.  13 

Nov.  9 

Means 

z  90000.  .  . 

_ 

_ 

1.  3998X10-" 

_ 

80000... 

— 

9.3418X10-13 

2.0280X10-10 

_ 

70000... 

*2.5231X10-20 

1.  4245X10-* 

1.1514X10-8 

— 

60000... 

*2.1316X10-10 

5.6128X10-8 

1.3990X10-7 

9.8014X10-8 

50000... 

*5.  1086X10" 

4.9474X10-7 

6.9750     " 

5.9613X10-7 

40000... 

3.3158X10-6 

3.  5993X10-" 

3.8849X10-6 

3.7423X10-6 

30000... 

1.3759X10-5 

1.4997X10-5 

1.5590X10-5 

1.4782X10-5 

20000.  .  . 

2.2020     " 

2.3711     " 

2.4864     " 

2.3532     " 

10000.  .  . 

4.0597     " 

4.4819     " 

3.8207     " 

4.12fO     " 

100.  .  . 

1.  1929X10-* 

1.  2171X10-* 

1.0703X10-4 

1.1601X10-4 

*  Omitted  from  the  means. 


This  pressure  probably  is  due  to  the  kinetic  movements  of 
the  electrons  in  the  volume  as  distinguished  from  P,  the  hydro- 
static pressure  in  the  gas  due  to  the  kinetic  energy  of  the  atoms 
and  molecules.  The  formulas  give  numerous  interrelations 
between  the  electronic  and  the  molecular  energies. 

Nichols  and  Hull  give  the  radiation  pressure   7.01  X  10"5 
erg.       dynes 
cm.3        cm.2 ' 

If  A  =  the  coefficient  of  absorption,  and  /  =  constant 
emissive  black  energy,  for  all  bodies  by  KirchhorTs  Law,  then 

(Tf  \ 

-j Ej  =  I  (1  —  A)  = 


ELEMENTS   OF  BLACK  RADIATION  IN  THE  ATMOSPHERES       177 


1     11         o  iS     o 

T-H     rHi-i::  :  r-i  :  :  FNI-C  1-1 
v    XX        X  XX    v 

COOOCOCOOJ  00  OMOOl 

C-COO5C-O  i-l  CMNCOI 

OOCOCCOOrJt  CO  COOrHI 

•tfOrHCMCO  -^  COCncO- 


o     ^jOrHCOco     •«     co*oo- 

rH         OOCO^t-r-t         iH         iHlHI 


i  I. 

x   x' 


0-2 

O-2 

0* 


X     XX  X 

10       T*lOTj<t-N       O       COOOrHOSOO  i-H 

SOOTliCOTHlO       CO       lOO5CMt-i-(  t- 

COt-COT-HCO       W       COOOOCDt-  t- 

r-|       CO^COt-OO       OJ       T-Hi-lfHCOCO  CM 


+ 


XXX 

lOCOO       r)<00-*OCO 


X10* 
X105 


>t-t-        00        00050>05rH        COCO 


xxxx 

•^•OO-^  U3 
OC-iH  t-  t 


._  ,_!»!<        COCOt>C~t-        CO        00  00  Ol  O5  O5 


1.1112X10- 

5.2515X10* 
2.5904X105 
7.7150  " 
1.6684X10* 


cocccoioeo     t- 

coTtcdt>od     ododododco"     oi 


xxx 

co  t-  mo 

Oi-lOCO 


XX    X 


•^COOiCOlO       CO       COOOC 

^SSSS?    S3    S5J 


:::::      3S2 

XXX 

COCOOOOOOO       OCOCO 

T-C^«t>OCO  «  t-  T^ 

co^ot-oco     ot^co 


t-t-t-0000       00       0000000000       OOOOOOOJOS        >-(COCO 


iiili  1. 

xxxxx   x" 


SSS§§     SSco^, 
oooc<i^t>*      coc^^^1^^ 

NCOCOCOCO        ir500^J«Ot*< 


:§ss 


1.  1. 


u 


IH  co  co  1-10     t-o-<*ooco     ^eo< 

OO  rH  CO  CO  CO    CMr-(O5t-CO    O  i-l  ( 
00  t-  rH  O5  00    rH  rH  i— (  rH  rH    rH  rH  i 


XX 

CO  iH  O  CO  CO  rH  O 

o  10  eo  eo  O5 1- 10 

rf  r-l  rH  00  1C  T)<  O5 

co  co  •*  •*  oo  05  eo 

,-J  rH*  ^-i  ^  CO*  CM  •*<" 


ooooo 
co^coooo 


ogoogoo 

o  o  o  o  o  o  o 

CO  T}<  CO  OO  O  CO  -^< 

1 1 1 1777 


178  A   TREATISE  ON  THE   SUN'S  RADIATION 

the  radiation  transmitted.    Hence, 
(148)  I  =  E  +  I  -AI 

and  the  emission  equals  the  absorption  for  any  other  gaseous 
medium.  Since  /  can  be  computed  from  u  in  all  thermodynamic 
atmospheres,  it  becomes  useful  in  discussing  the  side  of  the 
general  problem  that  pertains  to  the  electromagnetic  and  the 
electron  terms. 

TABLE  59 
VALUES  OF  /  IN  THE  EARTH'S  ATMOSPHERE 


1911 

June  9 

Sept.  13 

Nov.  9 

Means 

2  90000... 

_ 



1.  7590X10-" 

__ 

80000... 

_ 

1.1739X10-13 

2.5484X10-9 

_ 

70000... 

*3.1706X10'19 

1.7901X10-8 

1.4469X10-7 

— 

60000... 

*2.6786X10-9 

7.0533X10-7 

1.7580X10-' 

1.2633X10-6 

50000... 

*6.4196X10-7 

6.2171X10-6 

8.7650     " 

7.4911     " 

-40000... 

4.1667X10-5 

4.5234X10-5 

4.8819X10-5 

4.3240X10-5 

30000... 

1.7290X10'4 

1.8847X10-4 

1.9592X10-4 

1.8576X10-4 

20000... 

2.7671     " 

2.9795     " 

3.1246     " 

2.9571     " 

10000... 

5.1016     " 

5.6321     " 

4.8017     " 

5.1785     " 

100... 

1.4991X10-3 

1.5295X10-3 

1.3450X10-8 

1.4579X10-3 

*  Omitted  from  the  means. 

The  energy  of  black  radiation  on  the  photosphere  is  about 
the  same  as  in  the  earth's  atmosphere  at  40000  m.  The  energy 
at  ZR  goes  through  an  abrupt  change. 

p  =  the  dynamical  pressure  of  the  radiation  enclosed  in  a 
perfectly  reflecting  vessel;  F  =  the  pressure  exerted  mechani- 
cally by  the  electromagnetic  wave  front.  This  is  the  Maxwell 
pressure  of  light,  verified  by  Lebedew,  Nichols,  and  Hull.  These 

STT 
are  related,  F  —  —  p. 

The  equivalent  light  pressure  in  the  sun  to  that  in  the  earth's 
atmosphere  is  near  the  top  of  the  respective  gases.  This  pres- 
sure diminishes  very  rapidly  on  approaching  the  vanishing 
planes. 


ELEMENTS   OF  BLACK  RADIATION  IN  THE  ATMOSPHERES     179 


X10- 
X10-1 
" 


5.  6 
1.3 
2.7 
4.8 
8.2 


XXX    X 

o  —  inico 

§00  CM  U5  t- 
«  ^f  IO  C- 
iH  M  kO^  r-l 


S3 


i  4 


X 

«oo  eoto"*  t- 

tocooOTJico  o 

•^  OOTf  t-  CM  t- 

<O  t-  TH  oqOO  CO 

T-J  N  •«*  10  >0  «0 


i. 
XX" 


>co  x  eo  to     o» 


9356  XIO-17 
1202X10-" 
4754  XIO-13 


tOOiCOt-T-H       »O       OtOOlOO 
CO  ^»  •<*  Tj<  IO        IO        IO^«O«D«D 


16X10-6 
00X10-5 


XXX     X 


co  oo  t-eoo» 


XX 


!5«»«  » 

i  Tf^  to  to       IO 


1 

X 

IS  § 


t»  CO  T-H  IO  rH  TH 


coos  t-ioeo 


kOlOiOlO  to 


XX 

!2£    5SSS2 


>tDt*0904      iH 


XX     X 

03  to     Tj<  t-oeoeo 


ii. 

xxf 

CMCOCOCOCO        CO       TfTj«TjiTliTjt        lO«OOOOJr-l        CitOt- 

to' to  to  to  to     10     >ototototo 


xxxxx 

£88S§    S§?5^ 


•  CM  IO  IO< 


Tf  «>       t-05<Nrf  < 
T-HCO       OJONCO- 


xxxx 


t-t>000000       0000000000       00       0000000000       OOOOOOOJCft 


II 

x   x" 


ii 


i  Tt  05  to     i-H     c-eooi«o 


XX 

to  co  CM  90  o  c 


iHUSrHCO  U3 


•t>t>t-t-     t-t-t-t-t-     t- 


?o     «o  t-ooe 

t-'       t-'  t-'  C-  t-'  t-'       00  00  <3J  OS  i-i  i 


ooooo 

CMTl«OOOO 

1  1  II 


1      1 


180 


A  TREATISE   ON  THE   SUN'S   RADIATION 


TABLE  61 
VALUES  OF  F  IN  THE  EARTH'S  ATMOSPHERE 


1911 

June  9 

Sept.  13 

Nov.  9 

Means 

*  90000... 

_ 

_ 

1.  1727X10-" 



80000... 

_ 

7.8260X10-24 

1.  6990X10-" 

— 

70000... 

*2.1138X10-29 

1.1934X10-18 

9.6458X10-18 

— 

60000.  .  . 

*1.7858X10-19 

4.  7022X10-" 

1.1720X10-16 

8.2111X10-17 

50000... 

*4.2798X10-17 

4.1448X10-16 

5.8433     " 

4.9941X10-16 

40000.  .  . 

2.7778X10-15 

3.0157X10-15 

3.2546X10-15 

3.0160X10'15 

30000... 

1.1527X10-14 

1.2564X10-14 

1.3061X10-14 

1.2617X10-14 

20000... 

1.8448     " 

1.9864     " 

2.0830     " 

1.9714     " 

10000... 

3.4011     " 

3.7548     " 

3.2011     " 

3.4523     " 

100... 

9.9940     " 

1.0197X10'13 

8.9664    " 

9.7191     " 

*  Omitted  from  the  means. 


The  Nichols  and  Hull  value  of  p  gives  F  =  —  X  7.01  X  10~5 

c 

=  5.8727  X  10~14  and  this  occurs  near  the  top  of  the  isothermal 
layer  in  the  earth's  atmosphere. 

(dS\         4   u      4     aT* 
From  the  differential  equation,  \~Ty)T  =~^  ~T=~%  '  ~T~~~ 

4 

—  a  T3,  it  follows  that  the  specific  entropy  per  unit  volume  per 

o 

degree  is  5  =—  a  T3.     The  ratio  at  ZR  is  3.3012  X  10~7,  and  this 

o 

is  equivalent  to  using  a  temperature  7690°.2.     It  should  be  noted 

ds        1 
that  the  structure  of  the  spectrum  depending  upon  —  ==  j^ 

(Planck,  135)  refers  to  the  pure  aether  enclosed  with  matter  in 
an  adiabatic  volume,  and  it  is  not  directly  applicable  to  gaseous 
atmospheres  open  to  space,  wherein  the  term  for  work  d  W 
cannot  be  omitted,  and  d  W  =  P  d  v  assumed  negligible. 

8L       ds 
Planck  has  developed  two  cases  for  the  relations    -     =  -: 


(149)   I.  (135),          =         =     ;    II.  (81),  L  =  s. 


P,  and  (76) 


ELEMENTS   OF  BLACK  RADIATION  IN  THE  ATMOSPHERES      181 


i  |i, 5 . 

x  xx"  " 

•*J<  COiOl«Tj<00 

CO  O5COO3  CD  t- 

rji  CON  •*  COCO 

o  OOCO<NT}<CO 


5 
0X 

4 
2 
9 


SDr-l&lttiO         CO         CO  t~  i-M 


a  i. 

x  x" 

rH 

CO  t1- rH  T-H  t>  rH         CO 

O  COOr-NCO       04 


X 

>  OCOr-l        O 

I23S    £ 


XXX 


<N  i-!  i 


(M<M(MCOCO       CO        COCOCOTjt^t        ^H  < 


&u 

xxx"  x 


:      ,,«,,      i 

X 

CD        COCDlOlOlO         ^f  rH 
^         iOCOt5o004         OiH 


8 


li-H^CDrH        <N  (M  CO  CO  CO CO       CO  CO  CO  CO  CO 


^'od      S 

(MCDi-l(M       N 


i 

x"  " 

COOSiHOOO 

ooco  c<i  01-^ 

(MOlt-^HCD 

i-i  N  gj  co'  co' 


)0       N       NCOrjtl 

>O>       t-       lOCOi-H  < 


X 

»-l  O  t>CO 


03  CO  CO  CO  CO 


eo coco  co  co  co 


o     o 


XX  X 

oo  t-  ••*  oocoooeo 

HOOO  t-  OOO5  T-I  M 

coko  t-ooos  1-101 

CD  CO  CS1  IN  M  CO-CO 


x 

00  1ONO5COCO  0000000000  P5lOrJ« 

00  OOOOt-t-t-  IONO5COCO  NW^< 

CO  ^iOCOt-OO  CO  CO  CSI N  N  OOOiO 

CO  CO  CO  CO  CO  CO  ^  IO  CO  t~  00  W  N  CO 


i-i  IO  O  (M  L 

eg  co-«t  co 

00  OO  iH  ( 


xxxxx 

CO-*tiOt-OS  i-tCqiOOOi 

cocoooNio  Oicocoo< 

COCOCOOOO  t-lOCO,-l< 

CM  i-l  IO  •<*  OJ  OOOOiHi 

NCOCOCOi-l  TfrJtlOiOl 


iH  t>COO5U3       IO- 

tlNrHOiOO        rHI 


.11. .  I. 

'  xx"  "  x" 

iN^H       iH  00 -^  CO  N  00  00 
10SIO       00  U)  iH  TJI  OS  t-  CO 


CO  US  t-  OW 


IOIO  IfllOlO 


i.  o, 

x"  x" 


X 

iO  CO  00  O>  kO  Oi  ( 

C3  03  CO  00  OS  05  < 


NtO^CNlCO          "^Tj<T)<  T}l  • 


•*-*Tt<Tl<-5)<         Tj<         Tj<^Tj<Tj<Tj<         Tj<  Tj<  Tj<  Tl<  Tj«         kOlOlO  WrH. rH  T-H 


oo 

OOO 


ooo     o 


182 


A  TREATISE   ON  THE   SUN?S  RADIATION 


TABLE  63 
VALUES  OF  s  IN  THE  EARTH'S  ATMOSPHERE 


1911 

June  9 

Sept.  13 

Nov.  9 

Means 

2  90000.  .  . 

__ 

__ 

1.3997X10-26 

_ 

80000.  .  . 

— 

6.0859X10-15 

2.  0279X10-" 

— 

70000.  .  . 

*5.0461X10-20 

1.2662X10-10 

6.5791X10-10 

_ 

60000.  .  . 

*3.  2793X10-" 

2.8063X10-9 

5.5959X10-9 

4.2011X10-9 

50000.  .  . 

*3.5849X10-9 

1.7990X10-8 

2.3250X10-8 

2.0620X10-8 

40000.  .  . 

8.1366X10-8 

8.6736     " 

9.1948     " 

8.6683     " 

30000.  .  . 

2.5130X10-7 

2.6901X10-7 

2.7679X10-7 

2.6570X10^7 

20000.  .  . 

4.0890     " 

4.3148     " 

4.4559     " 

4.2866     " 

10000.  .  . 

7.1883     " 

7.6710     " 

6.9126     " 

7.2576     " 

100... 

1.6471X10-6 

1.6644X10-6 

1.5466X10-6 

1.6194X10-8 

*  Omitted  from  the  means. 


c  a 


~  ^  L       4  1 

K  =  ^  .  T*,  whence  T;  =  -  ~. 

4  7T  A  O    1 


(150) 

We  have  followed  Case  I. 

The  volume  density  of  the  energy  u  =  a  T*  when  multiplied 

by  —  gives  the  radiation  energy,  K  =  — .  a  T4,  as  a  flux  in 

one  direction.     Similarly,  the  volume  density  of  the  entropy 

4  c 

3  =  -  a  T3  when  multiplied  by  -j—  gives  the  entropy  radiation 

O  ~t  TT 

as  a  flux  L  =  —  .  a  T3. 

O  IT 

4 

The  Cases  I  and  II  differ  by  the  factor  -. 

O 

When  the  volume  density  of  the  electromagnetic  intensity  is 
multiplied  by  c,  the  velocity  of  propagation  in  the  aether,  3  X  1010, 
and  divided  by  J  d  Q  =  4  TT,  we  have  the  radiation  flux  in  %ne 
direction,  kinetic  energy  per  unit  volume  times  velocity,  which 

is  equivalent  to  [^  ^2  .  —  .  —  the  kinetic  energy  per  unit 
area  per  unit  time,  as  a  flux  across  the  unit  surface,  and  this  is 
equivalent  to  ^.  The  transformation  factor  from  (M.  K.  S.) 
to  (C.  G.  S.)  is  1000  for  K,  while  it  is  10  for  u.  There  need  be 


ELEMENTS   OF  BLACK  RADIATION  IN  THE  ATMOSPHERES       183 


-i: 


b       II 
g   MIE- 

^     II 

«o 


^ 

s 


I-    - 

X*  " 

O  O  oo  O  -^ 


COO^-I  kO 


^§00  00^ 

CM  CO  TJ-  10  CO 


XX 


o     "* 


>CM  rHCOCOOCM  >O  IOI 
)»-(  lOOOi-tTfOO  »-l  TJM 
CM  ^TMOtQlO  CO  C0< 


XX  X 

«OOOi-lO       t- 


icoco     cMt-eoooeo     oo 

I  NT*        Oi-ITfCOO       j-< 


iCMOCO 

t~O5O 


<OCOt>0000 


S-  1 

x"  x 

eo  tf  o 

!O£H  CM 


i-IVOOOOOO       lOOiOOl 
Ot-t-lOtM        TfiOiOCOC 


O  t— ^  rH  00  kO 
t-  000000 
CO  CO  CO  CO  CO  t— 


si 

XX 


t-c~  t-  1-  1- 


lOCN  O  OT*          TH 


T*     <ococot-t-     t-OtHcoio     t-     Oi-ieo- 
co     Oi-ccoiot-     ooooooo     o     ooot 

TflOtOlOiQ        tOkOiQiOtO        10        tOCOCQC 


IS- 

xx" 


'eo      ooooooo      ooooo      o      ooooc 


CO  X  00  iH  i-l  CM  CO 


t-       t-0000< 


•^^^5  s^sss  sssss  s  ssss: 


o     . 

1-H-     - 

X 

g^OOCC§21 
OO^OgNNN 


777 


1  1  1 


il 

X  X 

«M   rf 


CO 

ss 

i  a 


So1 
i  t 


i    . 

CM 


n 


Z2 


S5 
7« 


I      VO      g1 

:   :I 

:    :-H 


184 


A   TREATISE   ON  THE   SUN  S   RADIATION 


no  confusion  if  volume  energy  and  surface  flux  are  carefully 
distinguished. 

TABLE  65 
VALUES  OF  L  IN  THE  EARTH'S  ATMOSPHERE 


1911 

June  9 

Sept.  13 

Nov.  9 

Means 

2  90000... 

_ 

_ 

2.  5063  XlO-8 

__ 

80000.  .  . 

— 

1.1151X10-5 

3.6310X10-2 

— 

70000.  .  . 

*9.0352X10-U 

2.2672X10-1 

1.1780 

— 

60000.  .  . 

*5.8716X10-2 

5.0248 

1.0020X10 

7.5224 

50000... 

*6.4189 

3.2212X10 

4.1628     " 

3.6920X10 

.    40000.  .  . 

1.4569X102 

1.5530X102 

1.  6463  XlO2 

1.  5521X10* 

30000.  .  . 

4.4996     " 

4.8167     " 

4.9560     " 

4.7574     " 

20000.  .  . 

7.3215     " 

7.7258     " 

7.9785     " 

7.6753     " 

10000... 

1.2871X103 

1.3735X103 

1.2377X103 

1.  2994X10* 

100... 

2.9492     " 

2.9801     " 

2.7692     " 

2.8995     " 

*  Omitted  from  the  means. 


TABLE  67 
VALUES  OF  K  IN  THE  EARTH'S  ATMOSPHERE 


1911 

June  9 

Sept.  13 

Nov.  9 

Means 

2  90000 

1  0025  XlO"7 

• 

80000 

6  6903  X10~5 

1  .  4524 

70000... 

*  1.8070  XlO-10 

1.0202X10 

8.2462X10 

-  

60000... 

1.5267 

4.0198X102 

1.0020X103 

7.  0199  XlO2 

50000.  .  . 

*3.  6587  XlO2 

3.  5433  X  10s 

4.9954     " 

4.  2694  XlO3 

40000.  .  . 

2.3747X104 

2.5781X104 

2.  7823  XlO4 

2.  5784  XlO4 

30000.  .  . 

9.8540     " 

1.0741X105 

1.1166X109 

1.0392  XlO5 

20000.  .  . 

1.5771X105 

1.6981     " 

1.7808     " 

1.6853     " 

10000.  .  . 

1.4538     " 

3.2098     " 

2.7366     " 

2.4634     " 

100... 

4.2720     " 

4.3585     " 

7.6653     " 

5.4319     " 

*  Omitted  from  the  means. 

Table  69  gives  the  total  inner  kinetic  energy  per  unit  volume, 
and  it  is  the  same  as  Table  36.  We  have  confined  the  com- 
putations to  the  monatomic  elements,  and  so  have  avoided  the 
question  of  the  potential  energy  arising  from  atoms  and  mole- 
cules of  higher  complexity.  The  interrelation  of  the  potential 
and  the  kinetic  energies  in  respect  of  Maxwell's  First  and  Second 
Electromagnetic  Laws  still  constitutes  a  problem  beyond  the  range 


ELEMENTS  OF  BLACK  RADIATION  IN  THE  ATMOSPHERES       185 


o  oo 2®-  ® 

X  XX~  "  "  XX*  X 

CO  T-H  b- TJI  C<1  IO  rjl  OJ^COOSO 

i-i  b-iococoN  1-4  N^coM<t- 

UD  cob-b-iob-  eo  b-oob-coo 

OJ  OOiHCOr-lO  OO  "tfOr-IOOO 

U5  rJ<i-INTl<b-  OO  OS  ^  N  Tf  t-H 


o     o. 

x  x" 


s- 

XX* 


ooo 

XXX 

333    §SS^: 


I  II 

"  X  XX 

CM       »HO5b-lOW  b- O 

•<*        CMO5b-iO«  OOOO 

Tf       b-OiCMU500  b-OO 

b-       Oii-ITj'COOO  Or-l 


oSoo 


XXXX 

>0000010  OOCOCOOb- 

'IOTJIO5OS  Tf  O5  T!<  Oi  CO 

>C-000>10  OiONCOlO 


000.    O       O 

F^  T-(  T-HJ      TH  ^H 

XXX    X    X 


OOeO-^Tt  r)<       "* 


o      inosoo' 
oo      b-OTj<( 

b-       0>OO< 


00  O 

3       :  :   :   :   :       ;  ;  FNTHS  IH 

XX  X 

10       kOlo'ujuilO       10  co' CON  CM  rH 


23  2=:  =  : 

XX  X 

ON  CO  T-I  >-l  >H  »-l       COCOOb-' 

•*£  Tf  »HOOiONO5       CMO5COC<1< 

NN  b-COOb-CO       CMCOiOb-C 


000 

XXX 


IH  Ncorftioco  iHtMe 

CO  O5N»OOOi-l  t>OO( 

o  >-!  co  TJ<  10  b-  coco< 

CO  tOCOCOCOCO  t-00< 


12%    SS! 
!SS    §S 


xxxxx 


§CDOTJ<00  OCOOrJ<C 

COOOIMCO  O5O5OOC 

b-OOCSOT-l  rHiHCMC<JC 

co  co  co  t>  t>  b-'  b-^  i>  t>  i 


22-1— 

xx"  x" 

OJ       b-COO5iO»H       CM  b-  TJ<  T-H  -<t  O  T}< 

cq     co^'ocooo     b- o  ko  co  10  co  T}< 


2-  S- 
x^  x" 


i  co  o  cooi 
>oseo  ION 
Ib-b-  >-IN 


OOOJOOiH 

•^  -^  us  us  us 
co  co'  co'  co  co'     o  co  co  co  co 


:•  -j  -^<  1-1  ^  t>o  ( 

>IOO  b-OCO  b-< 

)TJ<  10  b-  cooo  co< 

JiOlO  IOCDCO  b-l 


-  -  :   22:  = 

xx" 

oo  o  N  n  oo  b-  oo 

b~  10  co  00  O5  OS  TT 


COCOCOCOCO       COCOCOCOCO 


§OO 
OOCO 


OOOOO 
(N^COOOO 

M  M 


OO 


- 
- 


186 


A  TREATISE   ON  THE   SUN'S  RADIATION 


F-3  II 

I  ,s 


I       00|<N 


g 


cs  •*  co  10  rH     T*     oo  «  o>  t-  -# 

CO  Tl<  CO  OO'  rH        rH        rH  M'  CO  CO  Tf« 


!=: 
X 


Sco      m 
CM       CO 

>rH  CO        IO 


eo  t»u3 

rH  OOCCt 
<  i-H  O  IO 


rH        t>OOrHrHrH        rH        rH  CO  N  CO  CO        IO 


272X10" 
7X101 
7  " 


CO         00         NlOOOrH^         rHO 

CO         Tj<         mt-OOOrH         COO 
TH        1-4        rHrHrHCON        CC i  IO 


»si,i  a...-,,  =  ,.,r,  =  =  =  =  =  s 

xxx        x  x 

COW5OU3CO        OCOCOOSCO  W       OCOCOOO^JI  OOOOCOCOOO  CO 

OO"5COrHUi        COCOCOCOCO  CO       OCDCOO5CO  WkOCOt~00  IO 

rHCOrH-«tod        rH  rH  rH  rH  rH  rH        rH  rH  rH  rH  rH  N  00  U5  CO*  00*  rH 

S.     ,     S 

x"  "  x* 

^  QO  iO  00  CO        CO  00  O  C^l  ^*  CO        00  O  CO  ^O  00  Ci  CO  00  *C  00  1C 

t-OiCOOW       COCO^^Tj*  Tj*        TfUSlClCO  COOiWCOO  t» 

II  I ; , . 

XX     X*  " 

o||,   I 

XXX*  X 

T-I^H^HrfrH   rHrHrHrHrH   rH  rH  rH  rH  rH  rH   rH  rH  rH  rH  r-l  rH  rH  rH  rH  rH  CO  N  CO  CO  Tji  Tf  in 
O        O 

rn:  ::rH     :  :  :  :  :       :   :   :  :  :  :       :   :  :   :   :  ;:::;  5  3  2  ;  3  5  S 
X           X 

CD  Tf  O5  to  t-        COrHOOiC-       COCOCO-^lO  CO       OOOrHCO^O  US-^COCOrH  lOCOOSOSCO- 

TJ<  CD  rH  os  ^     ooosoorH     cococococo  co     coeocoeoco  TCCOOOOCO  rHcoco«io< 

CO  O  O  ^*  O       CO  CO  ^  ^  ^       ^  Tj< -^ -^ -^  ^J<       -^  ^*  ^  ^  ^*  ^  ^*  ^  kO  IO  CO  00  O  W  CO  < 

§0000  o     ooooo  ooooo  ooooooo 

OOCO-^CO  CO-tfCOOOO  OOOOO  OOOOOOO 

ICOrHrH  '  '      '      '      If  ^  T  T  T  f  f  f 


ELEMENTS   OF   BLACK  RADIATION  IN   THE   ATMOSPHERES     187 


of  this  research.  The  data  here  presented,  from  the  thermody- 
namic  side,  must  be  taken  into  the  account,  especially  at  the 
solar  source  of  radiation. 

TABLE  69 
VALUES  OF  (Ui—Uo)  IN  THE  EARTH'S  ATMOSPHERE 


1911 

June  9 

Sept.  13 

Nov.  9 

Means 

90000  

2.  5798  XlO5 

80000 

3  1797  XlO6 

6  4850  XlO7 

70000  . 

*4  1555  X104 

2  2265  XlO8 

3.9119X108 

60000  

*1.6122X108 

1.0568  XlO9 

1.2344  XlO9 

1.1456  XlO9 

50000  

*1.  6014X10 

2.9581     " 

2.9880      ' 

2.9731     " 

40000  

6.1829     " 

6.4619     " 

6.0909      ' 

6.2452     " 

30000  

9.9355     " 

1.0291  XlO10 

9.6396      ' 

9.9554     " 

20000  

1.5560X109 

1.5992     " 

1.4883  XlO10 

1.5478  XlO10 

10000  

2.4574     " 

2.5023     " 

2.3182      ' 

2.4593     " 

100  

3.5843     " 

3.6189     " 

3.4247      ' 

3.5426     " 

*  Omitted  from  the  means. 

P  is  the  thermodynamic  hydrostatic  pressure  due  to  the 
kinetic  energy  of  the  molecules  per  unit  volume,  and  p  is  the 

rM  U     1  -, 

kinetic  energy  of  the  electrons  per  unit  volume,  ^         -  .  —    ,  so 

that  their  ratio  is  a  simple  number.  Within  the  solar  isothermal 
layer  the  ratio  is  not  far  from  the  velocity  of  light,  as  if  p  X  c  = 
P,  so  that  electrons  having  the  velocity  of  light  have  the  effect 
of  molecules  in  producing  volume  pressure.  This  value  of  the 
ratio  P/p  is  found  near  the  bottom  of  the  earth's  atmosphere. 
Below  the  plane  of  solar  radiation  the  ratio  drops,  and  p  ap- 
proaches P  in  value. 

Certain  Data  in  the  Radiation  Terms  Computed  Directly  from 
the  Formula  u  —  a  T*  and  Its  Dependents  for  Black  Radiation 

It  will  be  convenient  to  give  some  examples  of  the  values  of 
these  radiation  terms  as  derived  from  the  standard  value  of 
log  a  =  —15.86494.  We  have  selected  certain  temperatures  for 
the  computation  in  the  atmospheres  of  the  sun  and  the 
earth. 


188 


A  TREATISE   ON  THE   SUN'S   RADIATION 


- 

7.8722X109 

1.6899X10* 
.  .  7.7793X10» 

& 

o     o.  ... 

x   x"  " 

»*        O5CMCM  OS  ••* 
TH        TH  CM  CS1  (NCO 

10 

-  Ill- 

xxx" 

ooFH^-!ast> 

r-i    : 

& 

1     =  =  =  =  = 

X 

O      as«ot-ot- 
d      IO^CD  t-  as 

•*J<         TH  iH  C<1  COiH 

lO 

co 

OS 

S        S  « 

x"  xx" 

«o«o  OTH  uj 

ooooMOico 

0 

X 

0 
CO 

?! 

Nco 

00  

XX~ 

lOlOr-l        OSOOt-COlO 
THiHTJ*        -#IOCOC>00 

1 

L.  1768XlOio 
L.4044  " 
L.6319  " 
L.8595  " 
J.  0870X10io 

II 

XX 

£2 

gg 

t  ^    : 

.. 

S3 

ooS.  .       ..... 

NCOOSt-lO        IOOU5OIO 
•*#•*$  i-t  CQ  •*!*        K3COCOt-C~ 

i 

00 

9.0862X109 
1.0098X101° 
1.1110  " 
1.2121  " 
1.3133  ' 

III-  - 
xxx"  " 

co  TJ<  TJ*  as  10     co 

CO  00  lO  O  1O       t> 
COTfrltCOO       •* 

N  TH  00  10  -^         TH 

10  ^    : 

•H        '• 

-2 

|  

x"  " 

OOOTHiHCO       CO-^COOOO 

ooooujiH      ojcoo-<*as 

THU300COTj<         COOOrHCOlO 

COCO'T)*'»OCO     t^t>oooooo 

00 

s 
,  ^ 

asasas  THTH 

1-  III  1 

X~   XXX     X 

CMOWNCO       0 

^gSS       5 
THU5Ot>CM         TH 

T-!  TH  ri  oo  TH     as 

as     ; 

o  oo    : 

10  co     . 

1  M    : 

*• 

II     :  :  :  :  : 
XX 

1 

s 
0 
"  X 

i-4U5CiCO  t- 

III 

T  ^    i 

CJco     «ou5t-oooo     asasasasas 

OS 

Oi  Oi  Oi  Oi  rH 

M 

xxx" 

0}  CO  (M  00  U5        O  *O  Oi  "^  00       O  CO  ^O  00  O 
•ICCKOOO        t>OiTH^<£>        OOOOOOOOOi 

I 

O  W  CO  iO  CO 

oo,  .  .  , 

xx" 

T      : 

CO  04  O  C^  CO       -^  T}1  1C  ^O  W        M3  W  1C  UD  lO 

1O 

U3WCOCOCO 

«ococot-t-     as  T-I  TJ<  us  10  w  Tf 

0 

-l-ll-   - 

$ 

X 

THOOWOOO     ooor-icoT)*     inuiiomio 

0 

in 
o 

oeo  t-  oeo 
TH  t-eo  oco 

CM  CO  kO  t-OO 

X     XX 

l|j 

THrHWCOCO        lOCOCOCOCO        COCOCOCOCO 

0 

co  co  co  co  CD 

COCOt-t-t-         OOTHrH-,TH^lTH 

*    '• 

i 

IIP  p.  1 

0 

OOOOO 

d-^*  co  ooo 

11117 

ra-^fCOOOO        OOOOOOO 
I      I     I      I   iH        <N^COOOO(N^t 

I      I     I      1      |              i       i       i       i    t—4  ?H  tH 

1       1    1    1    1    ,    ,    , 

ZR  

Ratios  
Diatomic  hyd 

ELEMENTS   OF  BLACK  RADIATION  IN  THE  ATMOSPHERES     189 


TABLE  71 

p 

VALUES  OF  —  IN  THE  EARTH'S  ATMOSPHERE 
P 


1911 

June  9 

Sept.  13 

Nov.  9 

Means 

90000 

1.3090  XlO5 

80000  .. 

9.7693X10' 

1.8391  XlO8 

70000  

*1.1260X105 

1.5633  XlO7 

1.6298  XlO7 

60000.. 

*3.5267X107 

8.  7074  XlO7 

7.1630     " 

7.  9352  XlO7 

50000 

*7  6247  XlO8 

3  4838  XlO8 

3.  0639  XlO8 

3  2739  XlO8 

40000  . 

6  8820     " 

7.1593     " 

6.4362     " 

6.8258     " 

30000  
20000  
10000...    . 

8.5668     " 
2.  5289  XlO9 
6.6751     " 

8.6018     " 
2.  5027  XlO9 
6.2360     " 

7.9017     " 
2.  2279  XlO9 
6.7222     " 

8.3568     " 
2.  4198  XlO9 
6.5444     " 

100  

8.3906     " 

8.2370     " 

9.2638     " 

8.6305     " 

*  Omitted  from  the  means. 

TABLE  72 
THE  NORMAL  VALUES  OF  THE  BLACK  RADIATION  TERMS 


Radiation 
Terms 

SOLAR  ATMOSPHERE 

TERRESTRIAL  ATMOSPHERE 

Radiation 
Layer,  ZR 

Photosphere 

Sea 
Level 

20000  m. 

40000  m. 

Temperature,  T.  .  . 
Volume  density,  u. 
Volume  pressure,  p 
Energy    of    radia- 
tion, /  
Radiant  pressure,  F 
Specific  entropy  ,'5. 
Entropy  radiation, 
L  
Intensity  of  radia- 
tion, K  

7655° 
2.5162X10 
8.3872 

1.0540X10* 
7.0265  XlO-9 
4.3825  X10-* 

7.8470  X106 
6.0069  X10W 
2.5350X1013 

3.0102  X1Q6 

7687° 
2.5585X10 
8.5282 

1.0717X102 
7.1447X10-9 
4.4376X10-3 

7.9437  X10« 
6.1079  XlO10 
1.1697X10" 

7.2252X105 

286°.3 
4.9229  XlO-8 
1.6410X10-5 

2.0621  X  10-* 
1.3747  XlO-14 
2.2926  XlO-7 

4.1050  XlO2 
1.1753  XlO8 
1.2327X10* 

6.0837  XlO1" 

219°.5 
1.7008X10-5 
5.6694X10-6 

3.8261  XlO-s 
4.7497X10-15 
1.0332X10-7 

1.8499X102 
4.0604  XlO4 
5.3894X108 

1.0020  XlO10 

166°.0 
5.5639X10-6 
1.8546X10-6 

2.3306X10-5 
1.5502X10-15 
4.4689X10-8 

8.0017X10 
1.3283  XlO* 
2.1728X108 

1.3227  X10> 

Kinetic  energy, 
(Ui  -  Uo)  
Ratio  of  pressures, 
•P 

P 

Pressure,  P  .  .    . 

25247850 
0.0000014889 
2207720000 

6161900 
0.000000629 
1277040000 

998337 
0.0012152 
2870300 

56808 
0.00015823 
1636900 

24531 
0.00001696 
872613 

Density,  —  p  
tti 

Efficiency,  m  R.  .  .  . 

Mean  values  of  the  entire  system  were  taken  for  P9  p,  R,  T, 
and  the  standard  log  a  =  —  15.86494  in  Table  26.  These 
values  should  be  compared  with  those  given  on  Tables  54-71, 


190 


A  TREATISE   ON  1HE   SUN  S   RADIATION 


in  order  to  understand  the  departures  that  occur  in  the  thermo- 
dynamic  computed  values.  It  should  be  noted  that  it  is  the 
variation  in  the  coefficient  and  exponent  of  u  =  a  Ta  that  is 
primarily  responsible  for  these  changes. 


10     11      12     13     14     15     16     17     18 


-10  -9    -8    -7     -6     -5    -4    -3    T2     -1 


FIG.  21.     The  Great  Discontinuity  in  the  Radiation  Data  Occurring  at  the 
ZR  Levels.     Calcium  Vapor, 

P  and  ( Ui  —  C/0)  have  no  discontinuity  at  the  ZR  level ;  w,  p,  K,  /,  s,  L,  all 
suffer  a  per  saltum  change  at  that  level  where  the  radiation  is  generated. 


ELEMENTS   OF   BLACK  RADIATION  IN  THE  ATMOSPHERES     191 


The  Discontinuity  at  the  Levels  Where  the  Solar  Radiation  is 

Generated 

In  order  to  illustrate  the  discontinuity  in  several  terms  on 
the  levels  where  the  solar  radiation  originates,  a  diagram,  Fig. 
21,  has  been  constructed  for  Calcium  Vapor  on  a  composite 
scale,  such  that  the  ordinates  are  in  kilometers  from  the  photo- 
sphere and  the  abscissas  are  the  logarithm's  characteristics.  The 
hydrostatic  pressure  P  and  the  inner  kinetic  energy  (Ui  —  UQ) 
run  out  smoothly  across  this  level  ZR,  but  the  density  u,  pressure 
/>,  intensity  K,  energy  7,  entropy  s,  and  entropy  radiation  L, 
all  suffer  the  marked  discontinuity  which  has  been  computed 
and  illustrated  in  the  tables.  It  is  seen  that  while  the  thermo- 
dynamic  terms  P,  p,  R  do  not  go  through  such  a  discontinuity, 
all  the  terms  involved  in  the  generation  of  the  radiation  have 
been  affected  by  this  sudden  variation,  so  that  the  physical 
processes  must  be  different.  The  thermodynamic  terms  are 
probably  confined  to  the  kinetic  and  the  potential  energies  of 
the  molecules,  while  the  radiation  terms  apparently  are  to  be 
referred  to  some  change  in  the  structural  motions  of  the  elec- 
trons as  free  electric  charges. 


CHAPTER  VI 

Reconciliation  of  the  Data  Derived  from  the  Pyrheliometer 
and  the  Bolometer 

The  Poynting  Equation 

THE  resolution  of  the  problem  of  the  intensity  of  the  solar 
radiation  outside  of  the  earth's  atmosphere  depends  upon  the 
interpretation  that  ought  to  be  placed  upon  the  two  sides  of 
the  Poynting  surface-flux,  volume-density  equation, 

(66)  j- 

The  surface  flux  is  equal  to  the  volume  changes  plus  the 
waste.  The  observations  with  pyrheliometers  and  bolometers 
are  commonly  recorded  in  terms  of  the  surface-flux,  density 

~\JT    T  2 

per  aether-volume  times  the  velocity  of  propagation,  ~2  • 
Ti  *  T-  =  r-3'  so  that  conversions  from  the  (M.  K.  S.)  to  the 

JL/      I.          -L 

(C.  G.  S.)  system  require  the  factor  1000.  On  the  other  hand, 
the  thermodynamic  computations  are  properly  conducted  in 

ML2    I         M 
the   volume-density   form,    with   dimension      ^2    •  j^  =  y-™ 

so  that  the  conversion  factor  is  10.  As  already  stated,  Mr. 
Very  has  made  the  erroneous  criticism  that  the  conversion 
factor  for  the  volume-density  data  of  thermodynamics  should 
be  1000  instead  of  10,  thus  transferring  the  surface-flux  factor 
1000  to  the  volume-density,  and  so  violating  the  law  of  dimen- 
sions. While  the  data  of  the  two  sides  of  the  equation  must 
be  equivalent,  no  confusion  can  be  permitted  regarding  the  con- 
version of  computations  from  one  system  of  units  to  another. 

The  fundamental  difficulty  with  this  problem  in  practise  is 
that  the  solar  radiation  in  the  electromagnetic  form  of  energy 

192 


RECONCILIATION  OF  DATA  193 

is  transformed  in  the  earth's  atmosphere  into  equivalent  thermo- 
dynamic  forms,  but  these  are  so  different  in  their  manifestations 
that  it  is  not  easy  fully  to  identify  them.  The  pyrheliometer 
produces  data  which  are  very  unlike  the  data  derived  from  the 
bolometer,  and  each  of  these  differs  in  form  from  the  thermo- 
dynamic  effects  as  recorded  by  the  atmospheric  conditions. 
In  fact,  it  is  still  impractical  to  trace  these  transformations  in 
detail,  and  this  cannot  be  done  until  the  physics  of  the  trans- 
formation of  energy  in  gases  has  been  fully  established.  At  the 
present  time  it  is  necessary  to  determine  the  total  effects,  first 
as  a  series  of  summations,  and  then  proceed  to  the  details  so 
far  as  it  is  possible  to  do  so. 

It  should  be  noted  that  Mr.  Abbot,  and  many  other  ob- 
servers, are  approaching  the  general  problem  from  too  narrow 
a  point  of  view,  and  are  trying  to  reconcile  these  conflicting 
results,  as  heretofore  indicated,  by  an  overemphasis  upon  the 
complete  sufficiency  of  the  pyrheliometer  to  solve  the  problem. 
They  are  obliged  to  assume  that  the  sun  does  not  radiate  like  a 
black  body  at  its  high  temperature,  7655°,  in  order  to  account 
for  the  bolometer  discrepancy,  thus  debasing  its  4.00  calories 
down  to  about  2.00  calories.  Nor  do  they  take  into  account 
the  positive  thermodynamic  evidence  that  4.00  calories  are 
expended  in  the  earth's  atmosphere.  Mr.  Abbot's  first  as- 
sumption that  the  sun  does  not  radiate  at  black  body  efficiency, 
at  a  temperature  of  7655°,  is  disproved  by  the  solar  thermo- 
dynamics, summarized  in  the  preceding  chapters;  his  second 
assumption,  that  the  terrestrial  thermodynamics,  as  computed, 
does  not  require  consideration  is  based  only  upon  Mr.  Very's 
erroneous  statements  regarding  the  conversion  factor  from 
(M.  K.  S.)  to  (C.  G.  S.),  thus  placing  himself  in  conflict  with 
the  law  of  dimensions,  as  well  as  with  the  uniform  practice  of 
meteorologists  and  mathematical  physicists. 

Mr.  Abbot  summarizes  his  evidence  regarding  the  Intensity 
of  the  Solar  Radiation  outside  the  atmosphere,  Smithsonian 
Miscellaneous  Collections,  Volume  65,  Number  4,  1915,  and 
repeats  his  opinion  that  this  intensity  is  1.93  calories  per  sq. 
cm.  per  minute.  He  arrives  at  this  conclusion  by  the  original 


194  A   TREATISE   ON  THE   SUN'S   RADIATION 

Langley  Method  of  discussing  the  Bouguer  Formula  of  deple- 
tion, as  if  the  extrapolation  of  the  graph  (sec  z,  log  7)  could  be 
extended  from  sec  z  =  1.00,  in  the  zenith,  to  sec  z  =  0.00,  on 
the  outside  of  the  earth's  atmosphere.  The  last  step  is  non- 
mathematical,  and  as  a  physical  process  it  is  entirely  unlike 
those  preceding  it  (sec  z\ .  /i),  (sec  z2 .  72),  etc.,  so  that  it  is  in- 
competent to  integrate  from  the  station  to  the  top  of  the  at- 
mosphere through  the  numerous  unknown  physical  processes 
involved.  The  formula  7  =  70  p se<  z  is  competent  to  determine 
p  at  the  station,  but  not  to  trace  out  the  varying  values  of  p  in 
the  higher  strata;  it  is  competent  to  determine  the  ratio  7//0 
at  the  station,  but  not  to  discover  the  constituent  physical  terms 
that  make  up  7o  as  a  thermodynamic  effect,  from  the  parameter 
T  to  the  other  terms  of  the  general  law,  P  —  p  R  T  in  its  non- 
adiabatic  free  atmosphere  application.  He  supposes  that  by 
repeating  this  imperfect  extrapolation  at  many  stations,  and 
under  many  surface  conditions,  the  resulting  agreement  among 
them  becomes  proof  that  the  physical  result  is  complete.  This 
is  accompanied  by  a  general  evasion  of  many  other  facts  which 
are  clearly  in  contradiction  to  his  pyrheliometer  result.  On 
the  other  hand,  we  shall  be  able  to  point  out  that  there  are 
great  physical  tracts  in  thermodynamics  which  are  never 
recorded  by  the  pyrheliometer,  as  compared  with  the  full  effect 
of  the  solar  radiation,  and  thus  establish  our  thesis  that  this 
apparatus  registers  only  a  portion  of  the  incoming  solar  energy. 
His  "  New  Evidence"  consists  of  pyrheliometer  data,  regis- 
tered automatically  up  to  the  height  of  about  22000  meters  in 
balloon  ascensions,  and  this  is  a  welcome  and  important  addition 
to  our  available  data.  These  data  will  be  accepted  as  substan- 
tially correct,  and  adopted  into  our  reconstructed  system.  Mr. 
Fowle  has  published  the  results  of  his  investigation  into  the 
percentage  of  absorption  and  scattering  by  dry  air  and  aqueous 
vapor,  as  determined  from  the  bolometer  graphs  at  Washington, 
D.  C.,  Mt.  Wilson,  and  Mt. Whitney.  These  will  be  incorporated 
into  the  system  without  any  discussion.  We  shall  show  that  there 
are  three  decisive  thermodynamic  factors  which  must  still  be 
taken  into  the  account,  in  order  to  reconcile  the  pyrheliometer 


RECONCILIATION   OF  DATA  195 

data  with  the  bolometer  data,  and  both  of  these  with  the  ther- 
modynamic  data  of  the  terrestrial  and  the  solar  radiation.  It 
is  probable  that  the  scheme  here  explained  is  capable  of  great 
refinement  regarding  the  physical  processes  in  atmospheres,  and 
that  these  will  lead  to  important  contributions  to  the  mathe- 
matical analysis  of  the  theory  of  radiation  and  the  underlying 
constitution  of  matter.  It  should,  also,  be  noted  that  the  new 
high-level  pyrheliometer  data  fit  in  with  the  curve  which  had 
already  been  adopted  to  represent  this  term  from  the  sea  level 
to  the  top  of  the  atmosphere.  The  new  data,  therefore,  serve 
in  no  respect  to  modify  the  original  problem,  which  is  to  explain 
the  broad  discrepancy  between  the  result  by  the  pyrheliometer, 
2.00  calories,  and  the  result  by  the  bolometer  4.00  calories, 
without  denying  the  approximate  efficiency  of  the  sun  as  a 
black  radiator  at  the  temperature  of  7655°  and  5.85  calories. 


The  Cause  of  the  Change  from  the  Total  Solar  Radiation  5.85 

Calories  to  the  Effective  Radiation  3.98  Calories  Received 

on  the  Outer  Strata  of  the  Earths  Atmosphere 

It  has  been  shown  from  solar  thermodynamics  that  the 
radiation  originates  as  black,  and  at  an  intensity  equivalent  to 
5.85  calories  at  the  distance  of  the  earth.  Of  this  amount  it  is 
found  from  the  terrestrial  thermodynamics  that  3.98  calories 
are  received  on  the  outer  strata  of  the  earth's  atmosphere  and 
that  about  1.40  calories  reach  the  sea  level.  The  former  of 
these  depletions  is  solar  and  the  latter  is  terrestrial.  In  order 
to  explain  the  change  from  the  true  solar  to  the  effective  solar 
radiation,  we  shall  assume  the  data  of  Table  55,  Vol.  Ill,  Smith- 
sonian Institution,  which  contain  the  relative  intensity  of  the 
radiation  for  several  different  wave  lengths,  from  X  =  0.323  /*  to 
X  =  2.097  ju,  as  measured  by  the  bolometer  along  a  solar  radius 
in  terms  of  1.000  at  the  center  of  the  disk  and  0.000  at  the  limb. 
The  slit  of  the  bolometer  is  set  to  a  given  wave  length,  and  the 
disk  of  the  sun  in  drifting  across  it  produces  a  relative  intensity 
curve  with  maximum  at  the  center.  Fig.  22  contains  the  set  of 
relative  curves,  the  dotted  line  corresponding  with  0.470  /*>  which 


196 


A  TREATISE   ON  THE   SUN  S   RADIATION 


is  the  common  maximum  ordinate  of  the  system.  The  large 
dots  show  the  limits  of  the  cosine-ordinates,  which  express 
approximately  the  law  of  extinction  for  the  shortest  wave  length. 
The  coefficients  of  intensity  are  taken  as  the  ordinates  and  the 


1.000 


.900 


.800 


.700 


.10 


.200 


.100 


.10  .20  .30  .40  .50     .55     .60     .65     .70     .75    .80  |    g  .90  §  {§  Sjl.OO 

* 

FIG.  22.     The  Loss  of  Radiation  Between  the  Center  and  the  Limb  of  the 
Disk  of  the  Sun.     (Abbot's  Table  53,  Vol.  III). 

parts  of  the  radius  as  the  abscissas.  The  phenomenon  of  the 
diminution  of  the  brightness  of  the  sun's  disk  from  the  center 
to  the  limb  is  commonly  attributed  to  scattering,  and  the  figure 
shows  how  this  is  to  be  distributed  in  the  spectrum.  The  short 
waves  are  depleted  more  than  the  long  waves,  and  they  both 
are  increasingly  absorbed  in  some  function  of  the  path-length 
from  points  below  the  surface  to  points  of  escape  in  the  direc- 
tion of  the  earth.  This  complex  subject  will  be  resumed  in 
Chapter  VII. 


RECONCILIATION  OF  DATA 


197 


•S3AJno  aq^ 
Xq  '3uip3O3Jd  asoq^  UIQJJ  spBtu  'UIBJ3 

ip    B    UIOJJ     pa^BOS    SBAV  UUinjOD 


o  o  o  o  o  o  o 
t-  o  •*  o  Tf  eo  oo 
•<f  10  m  co  «o  t-  c~ 


TfTfkO»OU5U5O«O«O«Ot't^t~ 


co  T?  «o  os  T-H 


cviN-^fTfTjtioioeo^j^t- 

OSO5OSO5O5O5OSOSOSOJO5 


oooooo 


'  r^  ,-!  i-i  Ol 


^H  Oi  O  0>  00  N  O 
CO  t-  05  O  CO  50  t- 

^  o  eo  t-  t-  t-  t- 


200  rH 
t-  o 

i-H  t-  Tf 


OOOOOOiOO5O500N<OOOO5 

•^•<Jt,-ICOC^NO01fl-*O5eO 


c<ic<JC<iNcoeoeoeoeoeo 


iH  M  00  ••*  C<I  OO 

<£>  eo  t-  in  «>  10 

o  eo  •*  «o  oo  os 


eo  eo  03  eo  co  eo  eo  03 


soeococococococoeoeococoeoeoco 


o  eo  co  <-H  »-i  T)< 
oo  eo  in  oo  o  eo 
eo  -^  -^ji  ^<  10  uj 


198  A  TREATISE   ON  THE   SUN'S  RADIATION 

In  order  to  determine  the  effective  intensity  of  the  solar 
radiation  towards  the  earth,  it  is  necessary  to  integrate  the 
parts  contributed  by  each  wave  length  over  one  hemisphere  of 
the  sun.  To  do  this  the  data  of  Table  73,  Section  II,  are  com- 
puted as  follows:  Take  the  mean  value  of  the  intensity  for 
two  successive  points,  and  multiply  this  by  2?r  X  Pi2,  where  pi2 
is  the  mean  radius  of  the  two  points.  Example: 

Line  0.323  /*,  Pl  =  0.650    II  =  0.775 
P2  =  0.750    72  =  0.690 


Pi2=  0.700   712  =  0.733 
27T  pi2712  =  6.283  X  0.700  X  0.733  =  3.224. 

Section  II  contains  the  products  for  each  group,  so  that  the 
horizontal  sums  give  relative  numbers  of  the  effective  intensity 
of  the  several  spectrum  lines  over  the  hemisphere.  Had  there 
been  no  depletion,  the  same  hemisphere  would  have  given  an 
intensity  1.000  in  each  zone,  with  the  total  41.585,  while  the 
line  0.323  /*  gives  24.112,  so  that  the  factor  of  efficiency  is  0.580 
for  this  line.  The  last  column  in  II  gives  the  efficiency  factor 
for  each  of  the  selected  lines  in  the  spectrum.  Whatever  may 
be  the  original  solar  radiation,  the  hemispherical  action  of  the 
sun's  atmosphere  is  to  reduce  the  lines  in  these  proportions. 

We  have  shown  that  the  solar  radiation  originates  at  T  = 
7655°,  as  black  radiation,  and  the  coordinates  as  computed  by 
the  Wien-Planck  Formula  appear  in  Table  79.  They  can  be 
compared  there  with  the  energy  spectrum  curve  for  6950°,  5810°, 
5450°,  as  well  as  with  the  bolometer  ordinates  at  Washington, 
Mt.  Wilson,  Mt.  Whitney,  and  the  total  as  extrapolated  by 
Abbot  to  the  outer  limit  of  the  atmosphere.  The  ordinates  are 
given  in  relative  numbers,  but  for  intercomparison  they  are 
homogeneous,  and  for  reduction  to  calories  the  factor  1/20.9 
is  sufficient. 

We  will  interpolate  for  the  curves  7655°,  6950°,  total  bol- 
ometer, the  ordinates  corresponding  with  the  wave  lengths  herein 
adopted.  Those  for  the  7655°-curve  will  be  multiplied  by  the 
factors  of  efficiency,  and  the  product  is  to  be  compared  with  the 
ordinates  for  the  6950°  and  the  total  curves. 


RECONCILIATION  OF  DATA 


199 


TABLE  74 

COMPARISON  OF  THE  BLACK  SOLAR  INTENSITY  OF  RADIATION  AT  7655°  WITH 
THE  EFFECTIVE,  THE  THERMODYNAMIC  INTENSITY  AT  6950°,  AND  THE 
EXTRAPOLATED  BOLOMETER  TOTAL. 


A 

T 
7655° 

Factor 

Effective 
at  Sun 

T 
6950° 

Total 
Bolometer 

0.323  M  

9.790 

0.580 

5  678 

5  406 

1  500 

0  386     .             ... 

10  552 

0  589 

6  215 

6  348 

3  690 

0.433     

10  294 

0  633 

6  516 

6  514 

5  472 

0  456     . 

9  884 

0  661 

6  533 

6  454 

6  051 

0.481     

9.411 

0  679 

6  390 

6  268 

6  056 

0  501     . 

9  031 

0  690 

6  231 

6  116 

6  052 

0.534    

8.337 

0  709 

5  911 

5  946 

5  768 

0  604    . 

6  898 

0  738 

5  091 

4  959 

4  994 

0.670     

5.721 

0  762 

4  359 

4  217 

4  070 

0  699     . 

5  188 

0  770 

3  995 

3  664 

3  664 

0.866     

3  191 

0  806 

2  572* 

2  511 

2  403 

1  031 

1  994 

0  830 

1  655 

1  603 

1  536 

1.225     

1  199 

0  845 

1  013 

0  986 

0  985 

1.655  

0.459 

0.884 

0  406 

0  389 

0  466 

2.097    .. 

0  204 

0  899 

0  183 

0  176 

0  211 

A  comparison  of  the  effective  radiation  at  the  sun,  as  com- 
puted from  black  radiation  at  7655°,  with  the  black  radiation 
at  6950°,  shows  that  they  are  practically  identical  throughout  the 
spectrum.  This  means  that  the  solar  radiation,  which  is  equiva- 
lent to  5.85  calories  at  its  strata  of  origin  below  the  photosphere, 
is  effective  at  the  earth  to  the  amount  3.98  calories,  and  that 
about  1.87  calories  have  been  lost  by  the  hemispherical  scattering 
from  the  disk  and  by  absorption  in  the  superincumbent  strata. 

By  comparison  of  these  curves  with  the  Abbot  ordinates  as 
determined  bv  bolometer  observations,  and  extrapolated  to  the 


200 

limit  of  the  earth's  atmosphere,  it  appears  that  there  is  sub- 
stantial agreement  from  X  =  0.500  /*  to  the  end  of  the  spectrum 
in  the  long  waves,  but  that  there  is  a  great  depletion  at  the  end 
of  the  spectrum  in  the  short  waves.  This  is  in  accordance 
with  the  data  in  Table  83  and  Fig.  26.  The  depletion  in  the 
earth's  atmosphere  can  be  discussed  under  the  thermodynamic 
terms,  which  agree  with  the  bolometer  in  requiring  3.98  calories 
to  fall  upon  the  earth's  atmosphere.  Of  this  the  pyrheliometer 
measures  only  1.40  calories  at  the  sea  level,  extrapolated  to  1.940 
calories  on  the  vanishing  plane.  This  is  only  half  the  effective  solar 
constant  and  it  is  only  one-third  the  true  solar  constant.  The 
question  as  to  whether  the  sun  radiates  as  a  black  body  seems 
to  be  settled  in  the  affirmative.  The  physical  laws  regarding 
the  processes  of  absorption  and  transmission  in  each  atmosphere 
remain  to  be  studied,  but  it  is  evident  that  we  have  obtained 
the  P,  p,  R,  T,  in  all  the  strata  concerned  in  the  sun,  and  through- 
out those  at  the  earth,  so  that  the  laws  of  transmission,  refraction, 
and  numerous  other  functions  can  be  approached  with  suitable 
material  for  their  discussion. 

It  should  also  be  remembered  that  radiation  from  an  iso- 
thermal layer,  if  sufficiently  thick,  must  be  black  radiation.  It  has 
been  found  that  the  solar  gases  all  pass  through  such  deep  iso- 
thermal strata,  and  we  should,  therefore,  expect  the  solar  radia- 
tion to  be  black.  Furthermore,  it  is  found  from  the  terrestrial 
thermodynamics  that  about  3.98  calories  of  black  radiation  are 
actually  consumed  in  the  earth's  atmosphere. 

Summary  of  the  Terrestrial  Thermodynamic  Data 

It  will  be  convenient  at  this  point  to  summarize  the  results 
of  the  thermodynamic  data  as  computed,  and  reported  in  Bulletin 
No.  4,  0.  M.  A.,  1914,  or  in  the  Meteorological  Treatise,  1915. 
These  summaries  are  compiled  for  the  balloon  ascensions,  Uccle, 
June  9,  September  13,  and  November  9,  1911,  from  Tables  12-26 
of  the  bulletin.  The  sums  were  taken  for  each  ascension  through 
the  5000-meter  intervals  indicated  in  Table  75  under  z\  —  So; 
the  means  of  these  sums  in  each  stratum  for  the  three  ascensions 
were  taken  in  the  (M.  K.  S.)  system  of  the  computations,  and 


RECONCILIATION  OF  DATA  201 

these  were  converted  into  the  equivalent  gr.  cal./cm.2  min.  for 
the  following  discussions;  finally,  the  successive  sums  2  were 
computed  from  the  vanishing  plane  to  the  bottom  of  the  re- 
spective strata.  The  first  and  second  columns  under  each 
element  refer  in  the  first  to  the  strata-means,  and  in  the  second 
to  the  general  sum  from  the  top  of  the  atmosphere  to  the  bottom 
of  the  several  strata.  °  The  third  column  under  2  —  (Qi  —  Q0) , 
2  —  (Ui  —  Uo),  2  g  (zi  —  ZQ),  has  been  computed  by  subtracting 
1.5804  from  the  second  column,  for  reasons  that  will  be  stated. 
In  order  to  understand  the  relations  of  the  thermodynamic 
terms  to  one  another,  a  resume  of  the  non-adiabatic  formulas 
is  given: 

Resume  of  the  Thermodynamic  Formulas 

(isi)  _  A^Z»  =  +  (w1-  w,}  -  RIO  (r.  -  TO), 

Pio 

-  Cpw  (Ta-  TQ)  =  (Rw  -  Cpio  -  Rio)  (Ta-  T0). 
Pi-  Po  _ 
Pio 

-  Cpw  (Ta-   To)   =    (Cpa-  Cpio  -  Cpa)   (Ta~   T0). 

(152)  -  R10  (Ta-  To)  =  -  Pl""P°  -  (Wi  -  Wo), 

Pio 

—  R 10  (Ta —  TO)  =  (  —  Cpio  —  RIO  ~t~  Cpio)  (Ta —  TQ). 

-  RlO  (Ta~   TQ)=    +  g  (Zi  -  Zo)  +  (Ui  -   Uo), 

~  RlO  (Ta-   To)   =    (  ~  Cpa  +  Cpa  ~  Rio)   (Ta  ~  TO). 

(153)  +  (Wi  -  Wo)  =  -  Pl  "  P°  +  *io  (Ta-  To), 

Pio 
(Rio  —  Cpio)  (Ta—To)  =  (—  Cpio  +  RIO)  (Ta—  TO). 

+  (Wi  -  Wo)  =  -Pl~P°  -  (U,  -  Uo)-  g  (*i  -  zo), 
Pio 

Ta-To)   =    (-  Cpjo  -  Cpa+  RlO  +  Cpa)   (Ta~   TQ). 


(154)  -  (Ui  -  Uo)  =  +  #10  (Ta  -T0)-\-g  («i  -  zo), 

-(Cpa-Rw)  (Ta-To)   =    (Ru  -  Cpa)    (Ta-   TO). 

-  (Ui  -  Uo)  =  +Pl~P°+  (Wi  -  Wo)  +  g  (zi  -  zo), 
Pio 

-  (Cpa- Rio)  (Ta-To)  =  (Cpv>  +  .#10  -  Cpm  -  Cpa)  (Ta-To). 

(155)  -  (ft  -  Co)  =  -  (Wi  -  Wo)  -  (Ui  -  Uo), 

-To)  =  (-  Rio  +  Cpw  -  Cpa+  RIO)  (Ta-T0). 


202  A  TREATISE  ON  THE  SUN'S  RADIATION 

-  (ft  -  Co)  =  g  &  -  So)  +— — — 

PlO 

-(Cpa-Cpu)  (Ta-To)  =  (-  Cpa  +  Cpu)  (Ta~  T0). 

-  (Qi  -  ft)  =  g  fa  -  2b)  +  Pio  (Tfl-  To)  -  (Wi  -  Wo) , 

-  (Cpa-Cpw)  (T0-T0)  =  (-  C^«+  Pio  -  Pio  +  Q>10)  (Ta-  TQ). 


(156)  g  (Zi  -  Zo)  =    -  (ft  -  ft)  +  (Wi  -  Wo)-RlO  (Ta-  To), 

Pi-  Po 


g  (zi  -  Zo)    =  -  (ft  -  ft)  - 


PlO 


-  «b)  =  -  (IFi  -  TFo)  -  (tfi  -  Z70)  - 


Pio 


g  (21  -  Zb)   =    ~    (£^1  -    ?7o)  -  ^10  (Ta   ~   To), 

-  Q>a  (Ta~   To)  =    (-  Q>a  +  RlO  ~  RU)    (Ta~   TO). 

These  formulas  are  easily  verified  from  Fig.  23. 

Special  attention  may  be  directed  to  the  following  facts: 

1.  The  boundary  8  A  =  2  g  (zt  —  ZQ)   limits  the  adiabatic 

•p        -p 

system,  while  4  B  A  =  S  --  limits   the  non-adiabatic 

PlO 

system,  and  the  difference  between  them  is  S  —  (ft  —  ft)  = 
04C^4=4J5^48,  counted  in  the  proper  algebraic  directions. 

2.  The  work  2  +  (Wi  -  PF0)  is  the  area  03^=4C^7. 

3.  The  efficiency  S  —  ^10  (Ta—  T0)  =  3  ^1  5  4  =  7  ^L  8. 

4.  The  2  -  ^^L°  =  S  -  (ft  -  ft)  =  i  Sg  (*  -  20). 

PlO 

5.  Hence,  the  adiabatic  area  is  formed  from  the  non-adiabatic 
area  by  simply  adding  and  subtracting  Cpa  (Ta  —  To),  as  in 
formula   (4).     It  follows   that   the   differences   for   the   strata, 
(Wi  —  Wo),  (Ui  —  Uo),  (Q  i—  ft),  are  the  same  in  both  systems, 
and  that  we  can  transfer  the  area  4  D  A  8  to  coincide  with  the 
area  0  A  D  4,  and  this  is  convenient  for  several  reasons. 

p  _  p 

6.  The  total  hydrostatic  energy  2  =  —  -  -  -  measures  the 

Pio 

effect  of  the  solar  radiation,  after  it  has  been  transformed  from 
the  electromagnetic  form  into  the  thermodynamic  form,  and  it 


RECONCILIATION  OF  DATA 


203 


can  be  measured  by  the  sum  of  the  free  heat  as  well,  S  —  (Qi  —  Q0). 
The  sum  for  Abbot's  value  of  the  solar  constant  1.94  calories  is 
found  on  the  hydrostatic  curve  at  about  18000  meters  above  the 
sea  level,  on  the  free  heat  curve  at  40000  meters,  but  on  the  black 
radiation  curve  at  7000  meters.  Mr.  Abbot  has  taken  my 

TABLE  75 

SUMMARY  OF  THE  TERRESTRIAL  THERMODYNAMICS 
Ucde,  June  9,  Sept.  13,  Nov.  9,  1911,  Data  from  Bulletin  No.  4 


Pi-Po 

f 

2ff\          r\   \ 

z\  —zo 

2  (Wi  —  Wo) 

PlO 

(tfi     (Jo) 

70-66 

0.0184 

0.0184 

0.0254 

0.0254 

0.6623 

0.6623 

-0.9181 

65-61 

0.0321 

0.0505 

0.0439 

0.0693 

0.6441 

1.3064 

-0.2740 

60-56 

0.0572 

0.1077 

0.0780 

0.1473 

0:6118 

1.9182 

0.3278 

55-51 

0.0925 

0.2002 

0.1279 

0.2752 

0.5625 

2.4807 

0.9003 

50-46 

0.1386 

0.3388 

0.1926 

0.4678 

0.4990 

2.9797 

1.3993 

45-41 

0.1584 

0.4972 

0.2521 

0.7199 

0.4694 

3.4491 

1.8687 

40-36 

0.1410 

0.6382 

0.1988 

0.9187 

0.4955 

3.9446 

2.3642 

35-31 

0.1623 

0.8005 

0.2283 

1.1470 

0.4669 

4.4115 

2.8311 

30-26 

0.1993 

0.9998 

0.2804 

1.4274 

0.4157 

4.8272 

3.2468 

25-21 

0.2521 

1.2519 

0.3546 

1.7820 

0.3426 

5.1698 

3.5894 

20-16 

0.3183 

1.5702 

0.4496 

2.2314 

0,2467 

5.4165 

3.8361 

15-11 

0.3985 

1.9687 

0.5657 

2.7971 

0.1374 

5.5539 

3.9735 

10-  6 

0.4563 

2.4250 

0.6259 

3.4230 

0.0784 

5.6323 

4.0519 

5-  1 

0.4691 

2.8941 

0.6591 

4.0821 

0.0302 

5.6625 

4.0821 

2.8941 

4.0821 

5.6625 

31—  ZO 

2-(tfi-Z7o) 

2  g  (a  -a) 

70-66 

0.6807 

0.6807 

-0.8997 

0.6879 

0.6879 

-0.8925 

65-61 

0.6762 

1.3569 

-0.2235 

0.6890 

1.3769 

-0.2035 

60-56 

0.6679 

2.0248 

0.4444 

0.6901 

2.0670 

0.4866 

55-51 

0.6551 

2.6799 

1.0995 

0.6912 

2.7582 

1.1778 

50-46 

0.6362 

3.3161 

1.7357 

0.6924 

3.4506 

1.8702 

45-41 

0.6278 

3.9439 

2.3635 

0.6934 

4.1440 

2.5636 

40-36 

0.6364 

4.5803 

2.9999 

0.6945 

4.8385 

3.2581 

35-31 

0.6292 

5.2095 

3.6291 

0.6956 

5.5341 

3.9537 

30-26 

0.6152 

5.8247 

4.2443 

0.6967 

6.2308 

4.6504 

25-21 

0.5991 

6.4238 

4.8434 

0.6978 

6.9286 

5.3482 

20-16 

0.5647 

6.9885 

5.4081 

0.6989 

7.6275 

6.0371 

15-11 

0.5379 

7.5264 

5.9460 

0.7000 

8.3275 

6.7471 

10-  6 

0.5243 

8.0507 

6.4703 

0.7011 

9.0286 

7.4482 

5-   1 

0.4996 

8.5503 

6.9699 

0.6882 

9.7168 

8.1364 

8.5503 

9.7168 

204  A   TREATISE   ON  THE   SUN'S   RADIATION 

TABLE  75— Continued 


Zl—ZO 

2  Rio  (Ta  -To) 

2  A  J0 

2  A7a 

S-Ja 

70-66 

0.0070 

0.0072 

0.00039 

0.00039 

0.00002 

0.00002 

65-61 

0.0188 

0.0200 

0.00070 

0.00109 

0.00008 

0.00010 

3.990 

60-56 

0.0396 

0.0422 

0.00211 

0.00320 

0.00029 

0.00039 

3.990 

55-51 

0.0750 

0.0783 

0.00577 

0.00897 

0.00098 

0.00137 

3.989 

50-46 

0.1290 

0.1345 

0.01282 

0.02179 

0.00531 

0.00668 

3.983 

45-41 

0.2227 

0.2001 

0.08993 

0.11172 

0.00922 

0.01590 

3.974 

40-36 

0.2805 

0.2582 

0.22285 

0.33457 

0.02343 

0.03933 

3.951 

35-31 

0.3465 

0.3246 

0.12940 

0.46397 

0.02665 

0.06598 

3.934 

30-26 

0.4276 

0.4061 

0.13468 

0.59865 

0.04446 

0.11044 

3.880 

25-21 

0.5301 

0.5048 

0.13975 

0.73840 

0.06792 

0.17836 

3.812 

20-16 

0.6612 

0.6390 

0.16625 

0.90465 

0.10836 

0.28672 

3.703 

15-11 

0.8284 

0.8011 

0.50154 

1.40619 

0.19539 

0.48211 

3.508 

10-  6 

0.9980 

0.9779 

1  .  15080 

2.55699 

0.40720 

0.88931 

3.101 

5-  1 

1.1880 

1.1665 

1.38090 

3.93789 

0.56672 

1.45603 

2.534 

3.93789 

1  .  45603 

statements  regarding  the  summation  along  the  2  —  (Qi  —  Q0) 
curve  as  admission  that  the  pyrheliometer  measures  the  true 
solar  constant,  because  this  curve  has  its  half-sum  at  that  height, 
where  the  atmosphere  has  already  a  very  low  density.  On  the 
other  hand,  the  half-sum  for  the  hydrostatic  curve  is  at  18000 
meters,  and  for  the  black  radiation  it  is  at  8000  meters.  These 
complex  summations  must  be  carefully  studied  before  their  final 
interpretation  is  permitted.  The  transformation  between  the 
electromagnetic  radiation  and  the  thermodynamic  volume-effects 
involves  some  processes  and  terms  that  the  pyrheliometer  does 
not  register,  and  it  is  on  this  account  that  Mr.  Abbot's  value  of 
the  solar  constant  is  erroneous.  The  purpose  of  the  foregoing 
summary  is  to  indicate  that  we  can  pass  from  a  non-adiabatic 
system,  whose  boundary  curve  is  Cpw  (Ta  —  jT0),  into  a  strictly 
adiabatic  system,  whose  boundary  is  Cpa  (Ta  —  To)  —  — 
g  (zi  —  z0).  The  Boyle-Gay  Lussac  Law  in  free  atmospheres  refers 
to  the  non-adiabatic  (P.  p.  R.  7\),  which  become  adiabatic  when 
such  impressed  forces  as  gravitation  are  superposed  upon  them. 
The  mirror-enclosure,  within  which  the  radiation  formulas  are 
usually  developed,  does  not  exist  in  free  atmospheres,  but  in 
these  gases  the  enclosing  walls  are  removed  and  these  thermody- 


RECONCILIATION  OF  DATA 


205 


TABLE  76 

SUMMARY  OF  THE  ABSORBED  AND  THE  BLACK  BODY  RADIATIONS  AS  COMPUTED 
WITHIN  THE  TERRESTRIAL  ATMOSPHERE 

gr.  cal. 


I.     S  Ja  =  the  total  radiation  absorbed  =  2 


cm.2  min. 


z 
Meters 

Uccle 
3 

Omahar 
3 

Europe 
6 

U.S. 

7 

Tropics 
6 

Means 

S-Ja 

90000  

3.980 

Data  from 

80       

_ 

— 

_ 

— 

— 

_ 

3.980 

Tables  32-36, 

70       

_ 

_ 

_ 

_ 

— 

_ 

3.980 

Bulletin  No.  4 

60       .... 

— 

— 

— 

— 

— 

•  — 

3.980 

50000.... 

0.001 

— 

— 

— 

—     . 

0.001 

3.979 

45        .... 

0.007 

— 

— 

— 

— 

0.007 

3.973 

40       .... 

0.016 

— 

— 

— 

— 

0.016 

3.964 

35       

0.039 

— 

— 

— 

— 

0.039 

3.941 

30000  

0.065 

_ 

_ 

_ 

— 

0.065 

3.915 

ft  •'"'...-'. 

0.110 

_  • 

_ 

0.110 

_ 

0.110 

3.870 

20       

0.177 

0.178 

_ 

0.175 

_ 

0.177 

3.803 

15       

0.286 

0.293 

0.274 

0.278 

0.280 

0.282 

3.698 

10000  

0.481 

0.488 

0.460 

0.483 

0.516 

0.486 

3.494 

8 

0.623 

0.615 

0.609 

0.615 

0.676 

0.628 

3.352 

6 

0.784 

0.820 

0.788 

0.785 

0.852 

0.806 

3.174 

4         

0.982 

1.029 

0.974 

0.999 

1.023 

1.001 

2.979 

Mt.  Whitney 

2 

1.217 

1.325 

1.196 

1.225 

1.220 

1.237 

2.743 

Mt.  Wilson 

000   .... 

1.455 

1.624 

1.407 

1.429 

1.465 

1.476 

2.504 

Washington,  D.  C. 

II.    2  Jo  =  the  total  black  body  radiation  =  2  (ciT^—c0  TV) 


gr.  cal. 
cm.2  min. 


z 
Meters 

Uccle 
3 

Omaha 
3 

Europe 
6 

U.S. 
7 

Tropics 
6 

Means 

90000  

QA 

- 

- 

- 

- 

- 

- 

oU         .... 
70       













60       .... 

0.003 

— 

— 

— 

— 

0.003 

50000  

0.014 

_ 

_ 

_ 

_ 

0.014 

45       

0.032 

_ 

_ 

_ 

_ 

O.S32 

40       

0.117 

— 

— 

— 

— 

0.117 

As  the  result  of  all  the  avail- 

35      .... 

0.345 

— 

— 

— 

— 

0.345 

able  thermodynamic  data  the 

30000  

0.478 

— 

— 

— 

— 

0.478 

value  3.98  has  been    adopted 

25       .... 

0.618 

_ 

— 

0.618 

— 

0.618 

for  the  surface  summations  of 

20       .... 

0.760 

0.760 

_ 

0.766 

_ 

0.762 

the  intensity  of  the  solar  radia- 

15      .... 

0.937 

0.962 

0.915 

0.887 

0.940 

0.928 

tion. 

10000.... 

1.352 

1.272 

1.259 

1.331 

1.635 

1.370 

8 

1.714 

1.498 

1.673 

1.704 

2.101 

1.738 

6 

2.178 

2.010 

2.164 

2.200 

2,599 

2.230 

4         

2.675 

2.408 

2.655 

2.780 

3.062 

2.716 

2 

3.274 

2.967 

3.210 

3.391 

3.647 

3.298 

000  

3.808 

3.609 

3.743 

3.946 

4.338 

3.909 

namic  curves  take  their  place.  The  mirror- wall  condition  is 
reached  only  near  the  bottom  of  the  earth's  atmosphere,  or  in 
the  adiabatic  strata  of  the  solar  gases,  and  it  is  on  this  account 


206  A   TREATISE   ON  THE   SUN'S   RADIATION 

that  the  practical  transformation  problem  has  been  so  difficult. 
The  main  result  that  we  wish  to  note  follows:  THE  SOLAR  RADIATION 
ORIGINATES  AS  BLACK  RADIATION  AT  AN  EQUIVALENT  OF  5.85 
CALORIES;  IT  HAS  BEEN  FOUND  TO  BE  EQUIVALENT  TO  ABOUT 
3.98  CALORIES  IN  THE  EARTH'S  ATMOSPHERE;  THE  EVIDENCE  IS 
THAT  IT  IS  STILL  BLACK  RADIATION  AT  THE  LEVELS  50000-60000 
METERS,  AS  INDICATED  IN  THE  TABLES  54~70. 

These  data  are  taken  from  Tables  32-36,  Bui.  No. 4,  0.  M.  A., 
and  they  summarize  the  amount  of  the  absorbed  radiation 
S  Ja)  and  the  black  body  radiation  2  /o,  down  to  the  levels 
indicated  on  the  column  of  z  in  succession.  From  a  consideration 
of  all  the  available  data,  taking  account  of  the  season  of  the 
year  of  the  25  balloon  ascensions  as  computed,  and  the  latitude, 
it  has  been  thought  proper  to  assume  S  /0  =  3.980  as  the  effec- 
tive solar  radiation  at  the  distance  of  the  earth  on  the  vanishing 
plane  of  the  atmosphere.  The  column  (S  —  Ja)  shows  the 
progress  of  the  depletion  to  the  sea  level,  and  it  is  closely  in 
agreement  with  the  bolometer  results  at  the  stations  occupied. 
Compare  Meteorological  Treatise,  pages  380,  389. 

The  Thermodynamic  Data  in  the  lOOO-Meter  Levels 

In  Table  75  and  Fig.  23  the  thermodynamic  data  were  sum- 
marized in  the  5000-meter  strata,  and  in  the  deep  strata  from 
the  top  of  the  atmosphere  down  to  the  levels  indicated.  In 
Table  77  and  Fig.  24  Jhe  data  are  retained  in  the  1000-meter 

strata  for  +  (Wi-  TF0),  -  Pl  ""  P°,  -(Qi  -  Qo)  and  -(Ui-UQ). 
Pio 

The  summations  for  the  black  radiation  S  A  /0  and  for  selective 
radiation  S  A  Ja  are  repeated  on  Fig.  24.  There  are  two  points 
to  be  indicated  in  the  construction  of  Figs.  23  and  24.  In  the 
balloon  ascensions,  as  extended  to  70000,  80000,  90000  meters,  it 
is  evident  that  the  summations  may  be  carried  too  high  above 
the  effective  strata,  into  an  asymptotic  region,  to  represent 
properly  the  true  solar  radiation.  The  incoming  column  of  the 
electromagnetic  radiation  may  be  represented  as  contained  be- 
tween parallel  planes  on  a  fixed  scale;  at  some  levels  this  begins 


RECONCILIATION  OF  DATA 


207 


to  transform  when  in  contact  with  the  gaseous  media,  and  it  is 
this  effective  height  that  is  to  be  determined.  On  Fig.  23  the 
area  0  4  C  A  =  2  —  (Qi  —  Qo),  and  this  is  also  equal  to  the  area 
4  B  A  8,  whenever  the  curves  4  B  A  and  4  C  A  coincide  at  the 


70000 


60000 


50000 


.40000 


30000 


20000 


10000 


0123  A  56  78 

FIG.  23.     Summary  of  the  Terrestrial  Thermodynamic  Data. 

points  4  and  A.  If  the  area  4  C  A  8  is  extended  too  high  the 
points  4  and  A  diverge  along  the  two  axes,  and  this  is  due  to 
the  arbitrary  addition  of  +  Cpa  and  —  Cpa  to  the  non-adiabatic 
formula,  which  is  required  to  produce  the  adiabatic  formula. 
Hence,  by  making  the  curves  4  B  A  and  4  C  A  coincide  at  4 
and  A,  the  effective  height  of  the  atmosphere  as  a  thermo- 
dynamic  medium  is  determined.  In  this  case  it  is  about  60000 
meters  above  the  sea  level.  My  first  computations  for  Huron, 
September  1,  1910,  assumed  this  height  to  be  50000  meters;  but 
the  second  computations  extend  it  to  60000  meters,  with  a  very 
rarefied  coronal  strata  of  an  asymptotic  character  continuing 
up  to  90000  meters.  The  thermal  contents  of  this  coronal 


208 


A  TREATISE   ON  THE   SUN  S  RADIATION 


stratum  are  insignificant,  although  it  can  be  clearly  developed 
in  the  computations. 

Having  thus  shown  that  the  base  0—4  is  equal  to  the  base 
4—8,  the  first  being  strictly  non-adiabatic,  and  the  second  the 


70000 


60000 


50000 


.40000 


30000 


20000 


10000 


"0  0.50  1.00  1.50  2.00  2.50  3.00  3.50 

FIG.  24.     The  Thermodynamic  Data  in  the  Several  Strata  Having  a  Depth 

of  1000  Meters. 

addition  (Qi  —  Q0)  required  to  transform  it  into  the  adiabatic 
system  g  (zi  —  z0)  existing  in  the  earth's  atmosphere,  it  becomes 
proper  to  superpose  the  two  systems  upon  each  other  when 
dealing  with  the  individual  strata.  Furthermore,  since  the  in- 
coming solar  radiation  has  one  fixed  value  from  the  top  to  the 
bottom  of  the  earth's  atmosphere,  3.98  calories,  it  follows  that  this 
must  be  the  value  in  every  stratum,  in  whatever  manner  it  may 
become  subdivided  by  the  physical  processes  of  the  transformation. 
Since,  by  Poynting's  Law,  the  summation  in  every  stratum  must 
be  the  same  for  the  radiation-flux  and  for  the  thermodynamic 
volume-density  of  the  energy,  we  shall  adopt  the  same  vertical 


RECONCILIATION  OF  DATA 


209 


boundary  for  the  two  systems,  namely,  an  ordinate  at  3.98 
calories,  the  effective  value  of  the  intensity  of  the  solar  radiation 
at  the  vanishing  plane  of  the  earth's  atmosphere.  Since  the 
gravitation  energy  in  the  1000-meter  strata  is  9806,  (M.  K.  S.) 
system,  this  is  equivalent  to  0.1405  gr.  cal./cm.2  min.,  and  it 
would  be  a  proper  scale  for  use  with  the  thermodynamic  data. 
But  it  is  more  instructive  to  adopt  the  full  summation  value 
3.98,  to  which  the  factor  of  reduction  is  28.74 — that  is,  m,  the 
molecular  weight.  At  the  same  time  the  small  asymptotic 
region  above  60000  meters  has  been  omitted.  It  may  become 
possible  to  resume  the  computations  on  these  high  levels,  adapt- 
ing them  more  closely  to  a  definite  normal  height  for  the  atmo- 
sphere at  its  coronal  base  levels.  At  present  we  are  concerned 
with  detecting  the  several  elements  into  which  the  solar  radiation 
is  transformed  within  the  earth's  atmosphere. 

TABLE  77 
THE  THERMODYNAMIC  DATA  IN  THE  1000-METER  LEVELS 


EXTERN. 

IL  WORK 

HYDROS. 

PRESSURE 

FREE 

HEAT 

INNER 

ENERGY 

Height, 

in  Meters 

z 

/TIT-           TT/    \ 

m(Wi- 

Pi-Po 

«,Pl-Po 

-«h- 

-m  (Qi- 

-O/i- 

-m  (Ui- 

Wo) 

pio 

PIO 

Qo) 

Qo) 

Z/o) 

Uo) 

70000.    .  . 

0.0030 

0.0849 

0.0041 

0.1478 

0.1334 

3.9233 

0.1363 

3.9180 

65000.    .  . 

0.0050 

0.1441 

0.0056 

0.1580 

0.1307 

3.7554 

0.1357 

3.8997 

60000  .    .  . 

0.0090 

0.2577 

0.0124 

0.3557 

0.1207 

3.4669 

0.1344 

3.8619 

55000.    .  . 

0.0156 

0.4486 

0.0197 

0.5656 

0.1169 

3.3594 

0.1321 

3.7970 

50000.    .  . 

0.0238 

0.6848 

0.0378 

1.0861 

0.1052 

3.0232 

0.1293 

3.7166 

45000  .    .  . 

0.0323 

0.9294 

0.0453 

1.3006 

0.0932 

2.6786 

0.1255 

3.6071 

40000  .    .  . 

0.0286 

0.8231 

0.0408 

1  .  1734 

0.0979 

2.8128 

0.1265 

3.6360 

35000.    .  . 

0.0300 

0.8628 

0.0422 

1.2134 

0.0968 

2.7815 

0.1268 

3.6443 

30000.    .  . 

0.0367 

1.0534 

0.0515 

1.4804 

0.0858 

2.4663 

0.1245 

3.5772 

25000.    .  . 

0.0459 

1.3177 

0.0645 

1.8531 

0.0750 

2.1552 

0.1208 

3.4726 

20000.    .. 

0.0579 

1.6630 

0.0815 

2.3408 

0.0581 

1.6686 

0.1158 

3.3275 

15000.    .. 

0.0730 

2.0975 

0.1031 

2.9624 

0.0357 

1.0270 

0.1087 

3.1246 

10000.    .  . 

0.0870 

2.4992 

0.1219 

3.5030 

0.0192 

0.5525 

0.1062 

3.0518 

5000.    .. 

0.0901 

2.5879 

0.1304 

3.7476 

0.0099 

0.2869 

0.1027 

2.9523 

000.    .  . 

0.0991 

2.8476 

0.1392 

4.0013 

0.0000 

0.0000 

0.1007 

2.8953 

The  first  column  under  the  respective  thermodynamic  terms 
is  the  mean  value  for  the  three  balloon  ascensions  reduced  to  gr. 
cal./cm.2  min.  This  column  is  multiplied  by  m  =  28.74,  the 
adopted  molecular  weight  of  the  atmosphere.  There  is  a  margin 
of  inaccuracy  in  the  lowest  stratum,  between  the  sea  level  and 
the  height  of  the  station  Uccle,  which  is  100  meters,  because  it 


210  A  TREATISE   ON  THE   SUN'S  RADIATION 

is  not  exactly  known  how  well  the  adiabatic  assumption  which 
has  been  adopted  for  this  stratum  actually  holds  true.  In  our 
final  result  we  have  taken  m  (Wi  —  W0)  =  —  m  (Ui  —  Z70)  =  2.82. 
When  the  gas  is  in  contact  with  the  surface  of  the  earth  it  is 
probable  that  the  thermodynamic  terms  undergo  a  small  modifi- 
cation, which  should  be  more  carefully  examined.  An  especial 
research  will  be  required  to  determine  the  mean  values  of  all 
these  terms  in  the  first  1000  meters  above  the  surface  of  the 
stations,  where  they  are  to  be  applied. 

Resume  of  the  Preceding  Results 

When  the  results  of  the  preceding  investigations  have  been 
collected  together,  it  appears  that  there  are  twelve  lines  of  com- 
putation and  observation,  more  or  less  independent  of  one  an- 
other, which  converge  upon  3.98  gr.  cal./cm.2  min.  as  the  equiva- 
lent of  the  effective  intensity  of  the  solar  radiation  on  the  vanish- 
ing plane  of  the  earth's  atmosphere. 

1.  Solar  thermodynamics,  Table  28,  Table  73 =  3.90 

2.  Free  heat,  2  -  (Q1  -  Co),  Table  75 =  4.08 

p  p 

3.  Hydrostatic  pressure,  2 — ,  Table  75 =  4.08 

4.  Inner  energy  and  external  work,  2  [—  (Ui  —  UQ)  — 

(Wi  -  TFo)],  Table  73 ,:.    =  4.08 

5.  Black  radiation,  2  A  70,  Table  75 =  3.94 

6.  Free  heat  in  the  stratum,  -  m  (Qi  -  Co),  Table  77.  .=  3.92 

p  p 

7.  Hydrostatic  pressure  in  stratum,  —  m — , 

Table  77 =  4.00 

8.  Inner  energy  in  stratum,  -m(Ui-  J70),  Table  77 . .    =  3.92 

9.  External  work  and  gas  efficiency  ?  m  (Wi  —  Wo),  +  RIO 

(Ta  -  To),  Table  77 =  4.02 

10.  Kinetic  +  potential/energy  +  absorbed   radiation 

(7  +  II  +  2  A  Jo) =  3-90 

11.  Total  radiation  by  Table  79,  I  +  II  +  III  +  IV  + 

V  +  VI =3.98 

12.  The  data  of  the  pyrheliometer,  Table  83,  Fig.  26. ...    =  3.98 
Adopted  mean  of  the  twelve  processes =  3.98 


RECONCILIATION   OF  DATA  211 

The  concurrence  of  so  extensive  a  series  of  methods  of  dis- 
cussion in  fixing  the  solar  radiation  at  the  flux-intensity  of  3.98 
gr.  cal./cm.2  min.  at  the  distance  of  the  earth  justifies  the 
rejection  of  Abbot's  imperfect  method  of  extrapolation  by  the  Bou- 
guer  formula — that  is,  those  results  from  the  pyrheliometer  which 
reach  only  one-half  the  solar  constant  reduced  to  the  distance  of 
the  earth,  and  only  one-third  of  the  true  solar  constant  at  its  strata 
of  origin  below  the  photosphere. 

1.  True  solar  intensity  of  radiation 5.85  calories 

2.  Effective  solar  intensity  at  the  distance  of  the 

earth 3.98  " 

3.  Effective  intensity  by  the  bolometer  (indicated)  3.98  " 

4.  Effective  intensity  by  thermodynamics 3.98  " 

5.  Extrapolated  intensity  by  the  pyrheliometer 1.95  " 

6.  Intensity  at  the  sea  level  by  the  pyrheliometer  1.50  " 

The  Potential  Energy  of  the  Solar  Radiation  in  the  Sun's 
Atmosphere 

The  primary  fact  regarding  the  electromagnetic  solar  radia- 
tion is  that  half  of  its  energy  is  kinetic  or  magnetic,  and  half 
of  its  energy  is  potential  or  electric.  This  holds  true  in  the 
aether,  in  the  space  between  the  atmosphere  of  the  sun  and  the 
atmosphere  of  the  earth,  but  it  is  not  true  in  the  gaseous  media 
of  either  of  these  atmospheres.  The  reason  that  the  kinetic 
and  the  potential  energies  are  not  equally  divided  in  gaseous 
media,  as  they  are  in  the  pure  aether,  is  that  gases  have  three 
degrees  of  freedom,  while  the  <zther  has  only  two  degrees  of  freedom. 
For  instance,  Planck  in  the  "  Theorie  der  Warmestrahlung," 
1913,  discusses  his  theory  of  the  Wirkungsquantum  as  the 
effect  of  electromagnetic  oscillations,  such  that  two  degrees  of 
freedom  are  sufficient  to  account  for  the  energy  that  is  in  opera- 
tion. The  equilibrium  of  electromagnetic  energy  is  always  due 
to  a  magnetic  induction  B  and  an  electric  displacement  D, 
which  are  related  by  (44)  to  (59).  Heaviside  gives  several  illus- 
trations of  these  energies,  when  an  electric  charge  is  suddenly 
moved  or  suddenly  stopped,  as  in  collisions,  with  the  conse- 


212  A   TREATISE   ON  THE   SUN'S  RADIATION 

quent  plane  electromagnetic  wave  projected  with  the  velocity 
of  light  v,  as  determined  by  the  induction  coefficients  of  the 
magnetic  field  /*  and  the  electric  field  K,  in  the  formula  ju  K  v2  =  1 . 
For  oscillators  there  are  only  two  variables  (/.  \I/) ,  where  /  =  the 
moment  of  the  electric  charge  =  the  product  of  the  electric 
charge  times  its  polar  distance  =  e+  a;  while  \fr  =  Lf,  the 
impulse  or  acceleration,  so  that, 


df .  d  $  =  h  =  constant, 


which  is  the  quantum  hypothesis.  On  the  other  hand,  the 
energy  of  a  moved  molecule  has  three  variables,  x  .  y  .  z,  and 
temperature  depends  upon  the  translation  and  collisions  of  the 
molecules  only,  and  not  upon  the  inner  structure  of  the  atoms. 
Let  U  =  the  total  energy  of  N  oscillators  in  the  unit  mass,  and 

hv 

-T-  the  mean  energy  of  the  oscillators  in  one  elementary  region, 

a 

having  a  frequency  v  for  the  electromagnetic  oscillations.  Thence, 
Planck  deduces  his  formula, 


(157)    U  = 


1  -  e  kr  CkT  - 

which  is  (87).     If  T  =  0,  no  temperature,  we  have, 

(158)  U  =  N  .  ~  (and  not  0), 

a 

hv 
so  thai  the  oscillators  send  out  the  mean  energy  —  even  when 

40 

the  temperature  T  =  0.  In  the  inner  part  of  the  atom  there  is 
energy  independent  of  the  temperature.  If  T  =  °°  ,  Ui  =  k  N  T, 
which  is  proportional  to  T  and  independent  of  k,  that  is,  of 
the  nature  of  the  oscillators,  namely,  the  electron-structure  of 
the  atoms  and  the  monatomic  molecules  themselves.  From 
(28)  we  have  the  total  inner  kinetic  energy  in  the  volume  V 
for  the  total  thermodynamic  inner  energy, 


RECONCILIATION  OF  DATA  213 

(159)  (Ui  -  Uo)  =  -|  k  N  T  =  Cv  m  T  =  Cv  P  T  V  = 

z 

|  P  7  =  -|  KT  =  HV, 
2  2i 

where  U0  is  some  initial  value  due  to  the  integration.  For  the 
unit  volume  V  =  1,  and  UQ  =  0,  we  have, 

o 

(160)  f/2  =  -^  k  N  T  for  thermodynamic  collisions, 

2i 

(161)  Ui  =  k  N  T  for  electromagnetic  oscillations. 

2 

Hence,  £/i  (for  oscillators)  =  —  Z72  (for  monatomic  molecules). 

o 

The  energy  of  electromagnetic  radiation  relative  to  the  energy 
of  monatomic  molecular  temperature  collisions  stands  in  the 
relation  of  3  to  2.  For  diatomic  and  triatomic  molecules  the 
ratio  is  different. 

It  will  be  remembered  that  the  solar  thermodynamic  com- 
putations were  confined  to  monatomic  elements,  in  which  it 
was  assumed  that  the  inner  energy  is  wholly  kinetic  and  that 
the  potential  energy  is  negligible.  Under  these  circumstances 
the  equivalent  solar  intensity  of  radiation  in  the  strata  of  origin 
ZR  at  certain  distances  below  the  photosphere  amounts  to  5.85 
calories.  On  its  arrival  at  the  outer  strata  of  the  earth's  atmos- 
phere it  has  been  found  to  be  reduced  to  3.98  calories.  These 
ratios  are  nearly  the  same, 

5.85       _3_       solar  intensity  (electromagnetic) 
3.98  ~    2    ~  solar  intensity  (thermodynamic) 

Since  the  solar  radiation,  originating  in  the  planes  ZR,  passes 
through  the  deep  strata  of  the  other  chemical  elements  which 
are  not  monatomic,  we  may  assume  that  one- third  of  the  original 
radiation  energy  in  the'  electromagnetic  form  has  'gone  over 
into  building  the  potential  energy  in  the  molecules  and  atoms 
of  higher  complexity.  This,  it  appears,  is  a  very  complete  ac- 
count of  the  first  great  depletion  of  the  energy  of  the  solar 
radiation  as  recorded  by  the  bolometer,  and  illustrated  in  Fig. 
25.  The  spectrum  of  the  bolometer  shows  that  the  long  waves, 
1.50  ju  to  2.50  fjL,  emanate  from  the  high  temperature  7655°, 


214  A   TREATISE   ON  THE   SUN'S   RADIATION 

and  succeed  in  passing  through  the  gases  of  both  the  solar 
and  the  terrestrial  atmospheres  without  depletion,  while  the 
shorter  waves  0.0  ju  to  1.50  ju  undergo  losses  such  that  their 
equivalent  temperature  is  6950°  and  their  efficiency  3.98  cal- 
ories. The  entire  history  of  these  depletions  of  the  short  wave 
lengths  in  the  solar  atmosphere  will  prove  to  be  of  value  in 
molecular  physics.  We  conclude  that  the  potential  energy  of 
the  gases  in  the  outer  levels  of  the  sun's  atmosphere  is  built 
up  by  the  average  expenditure  of  1.87  gr.  cal./cm.2  min.,  as 
a  flux  which  is  equivalent  by  Poynting's  Law  to  the  same  amount 
of  volume-density.  This  account  of  the  potential  energy  is 
in  conformity  with  the  depletion  by  scattering  already  men- 
tioned. 

The  Potential  Energy  of  the  Solar  Radiation  in  the  Earth's 
Atmosphere 

There  is  yet  another  loss  of  the  solar  radiation  due  to  the 
transformation  of  a  portion  of  it  in  building  up  the  potential 
energy  of  the  atoms  and  the  molecules  in  the  earth's  atmosphere. 
At  the  vanishing  plane  of  the  earth's  atmosphere  the  electro- 
magnetic energy  is  divided  equally  between  the  kinetic  (mag- 
netic) and  the  potential  (electric)  forms.  On  the  other  hand, 
the  gases  of  the  atmosphere  are  distributed  in  another  ratio, 
such  that, 

the  total  inner  energy,  U  =  1.641  =  Cv  p  T 

3 
the  kinetic  energy,        H  =  1.000  =  —  P 

the  potential  energy,     /  =  0.641  =  £/  —  H 
This  result  is  derived  from  the  general  formulas  of  Table  2. 


U        C*»T        2C^T        2Cp-i641 
H          \P      '   3  RpT"   3    R  ' 

(163)      U  =  H  +  J,  and  /  =  Z7  -  #  =  0.641. 

Examples  of  this  fundamental  relation  are  given  in  the  Meteor- 
ological Treatise,  Table  97,  and  in  the  following  Table  78: 


RECONCILIATION  OF  DATA 


215 


TABLE  78 
THE  TOTAL  INNER  ENERGY  U,  KINETIC  ENERGY  H,  AND  POTENTIAL  ENERGY  / 


0 

KINETIC  ENERGY  H  =  -$  P 

TOTAL  ENERGY  U  =  Cv  pT 

J  =  U  -  H 

1911 

June  9 

Sept.  13 

Nov.  9 

June  9 

Sept.  13 

Nov.  9 

z 
70000.  . 

3.971X10-21 

3.340X10-3 

0.02815 

6.515X10-21 

5.481X10-3 

0.4618 

65000.  . 

7.629X10-8 

0.07220 

0.2450 

1.252X10-7 

0.11847 

0.4019 

60000.  . 

1.128X10^ 

0.9229 

1.5031 

1.849X10-^ 

1  .  2027 

2.4661 

The   constant 

55000.  . 

0.1835 

5.006 

7.4326 

0.3011 

8.215 

12.194 

u 

50000.  . 

5.843 

25.794 

32.056 

9.588 

42.424 

52.591 

ratio  —  =  1.641 

45000.  . 

75.692 

111.76 

120.12 

124.21 

183  .  39 

197.07 

H 

40000.  . 

342.28 

386.56 

375.06 

561  .  07 

634.31 

615.34 

is  satisfied,  and 

35000.  . 

809.43 

898.54 

863.76 

1328.4 

1474.4 

1417.3 

it  is  a  check  on 

30000.  . 

1768.0 

1935.1 

1847.9 

2901  .  5 

3174.5 

3032.0 

the     series     of 

25000.  . 

3821  .  6 

4133.0 

3899.0 

6271.4 

6782.0 

6397.2 

computations. 

20000.  . 

8353.0 

8901.0 

8309.2 

13708. 

14607. 

13635. 

15000.  . 

18424. 

19455. 

17826. 

30236. 

31927. 

29251. 

10000.  . 

40649. 

41924. 

38528. 

66705. 

68803. 

63226. 

5000.  . 

82007. 

82710. 

79260. 

134570. 

136050. 

130065. 

100.. 

150140. 

150730. 

148720. 

246390. 

246780. 

244060. 

The  kinetic  energy  is  used  in  building  up  the  temperature  of 
the  gases  of  the  atmosphere,  which  is  due  to  the  transportation 
and  collision  of  the  molecules;  the  potential  energy  is  used  in 
storing  up  the  static  energy,  which  is  held  in  the  structural 
configuration  of  the  atoms,  the  molecules,  and  the  electrons, 
and  the  ions  of  which  they  may  be  composed.  This  change 
of  the  potential  (electric)  energy  of  the  electromagnetic  solar 
radiation  into  0.641  parts  of  the  gaseous  potential  energy  is 
shown  on  Figs.  25,  26,  where  the  region  I  is  the  kinetic  energy 
H  as  determined  by  the  pyrheliometer,  while  the  region  II  is  the 
potential  energy  7,  which  is  0.641  H.  It  is  supposed  that  the 
transformation  from  J  of  the  electromagnetic  field  to  /  of  the 
gaseous  field,  that  is,  from  1.000  to  0.641,  may  be  somewhat 
gradual  in  the  high  coronal  strata,  as  indicated  by  the  dotted 
line.  If  this  is  actually  the  case,  there  is  evidently  a  great 
production  of  free  electric  charges  in  region  VI,  where  the  elec- 
tricity, which  manifests  itself  as  auroral  discharges  and  mag- 
netic disturbances  of  the  earth's  normal  field,  is  to  be  located 
in  the  first  instance. 


216  A   TREATISE   ON  THE   SUN*S   RADIATION 

The  Third  Thermodynamic  Depletion 

It  yet  remains  to  be  noted  that  there  is  another  thermo- 
dynamic  term  which  must  be  built  up  out  of  the  solar  radia- 
tion before  an  equilibrium  can  be  obtained.  That  is  the  trans- 
formation of  Uv  =  Cv  p  Tj  the  total  inner  energy  which  is  de- 
pendent upon  the  specific  heat  at  constant  volume  C^io,  into 
the  specific  heat  at  constant  pressure  Cpa  upon  which  the 
gravity  effect  depends, 

(164)  Up  =  g  (*  -*>)»-  Cpa  (Ta  -  To). 

Since  Cpa  =  Cvw  +  RW,  this  is  equivalent  to  the  addition  of 
the  term  .Rio,  whose  values  may  be  studied  in  the  formula, 

(165)  +  R10  (Ta  -  To)  =  +  ^^°  +  OFi  -  Wo)  = 

PlO 

-(Ci-eo)-g(*i-*>). 

The  region  -  R1Q  (Ta  -  TQ)  is  indicated  as  III  on  Fig.  26,  in 
conformity  with  the  conditions  already  explained.  Compare 
Table  75. 

The  Scattered  and  the  Absorbed  Radiation 

The  incoming  radiation  suffers  two  other  important  deple- 
tions, (l)  by  scattering  or  reflection  on  the  molecules,  and  dust 
contents  of  the  atmosphere,  whereby  no  influence  is  effected 
upon  the  temperature,  the  energy  being  lost  to  space;  and  (2) 
by  absorption,  which  is  generally  a  potential  effect  rather  than 
a  kinetic  effect  upon  the  translation  of  the  molecules.  The 
region  of  scattered  radiation  is  numbered  IV  on  Figs.  25,  26, 
and  one  should  distinguish  between  the  low-level  scattering 
and  the  high-level  scattering,  the  former  probably  pertaining 
to  the  wave  lengths  0.35  /-i  to  0.80  M,  and  the  latter  to  the  shorter 
wave  lengths  0.00  n  to  0.35  M-  There  are  two  regions  of  ab- 
sorption, the  low-level  marked  V,  and  the  high-level  included 
in  III,  the  latter  above  the  isothermal  levels  wherein  there 
is  no  absorption,  and  the  former  below  the  isothermal  strata, 
extending  through  the  convection  region  to  the  sea  level. 

The  foregoing  analysis  admits  that, 


RECONCILIATION  OF  DATA  217 

(1)  The  pyrheliometer  measures  the  kinetic  energy  as  de- 
pleted by  the  scattering  and  the  absorption  in  the  lower  levels, 
that  is,  I,  IV,  V;  but  it  denies  that, 

(2)  The   pyrheliometer   measures   the   potential   energy   of 
region  II,  the  thermodynamic  term  R  of  region  III,  the  high- 
level  scattering  of  region  VI,  and  the  high-level  absorption  of 
region  VI,  as  well  as  the  ionization  of  the  same  region  VI. 
The  boundaries  within  VI  are  still  a  subject  of  research. 

The  Pyrheliometer  and  Bolometer  Observations 

We  shall  now  collect  together  the  accepted  data  of  the 
observations  derived  from  the  pyrheliometer  and  the  bolometer, 
in  order  to  learn  to  what  extent  they  conform  to  the  foregoing 
analysis  of  the  thermodynamic  data.  It  will  be  remembered 
that  there  are  two  distinct  methods  of  discussing  the  pyrheli- 
ometer observations  heretofore  employed: 

(1)  The  Langley-Abbot  method,  in  which  only  the  kinetic 
energy,  as  it  is  effective  at  the  instrument,  is  accounted  for;  and, 

(2)  The  Bigelow  method,  which  is  competent  to  include  the 
potential  energy  with  the  kinetic  energy,  at  least  to  some  extent. 
The  former  method  of  extrapolation  results  in  a  series  of  values 
that  are  inconsistent  at   the  several  stations  when  taken  in 
sufficient  detail.     The  latter  harmonizes  them  in  a  form  which 
is  simple  for  intercomparisons.     Neither  of  these  pyrheliom- 
eter methods  succeeds  in  registering  all  the  terms  which  are 
demanded  by  the  bolometer  and  by  the  thermodynamics  of  the 
earth's  atmosphere.     This  is  in  the  Meteorological  Treatise. 

(3)  A  third  method  of  reducing  the  pyrheliometer  observa- 
tions to  about  3.98  calories  is  illustrated  in  Table  83,  and  all 
our  data  have  finally  been  recomputed  by  this  method,  since 
it  comprises  the  terms  I,  II,  III,  IV,  V,  VI,  and  requires  no 
extrapolation  beyond  sec  z  =  1,  thus  being  complete  on  the 
level  of  observation. 

The  general  view  is  that  while  the  electromagnetic  radiation 
has  the  kinetic  and  the  potential  energies  in  equal  parts,  and 
the  atmosphere  has  1.00  part  kinetic  and  0.641  part  potential, 
the  complex  silver  disk  of  the  pyrheliometer  turns  1.00  part 


218  A   TREATISE   ON  THE   SUN'S  RADIATION 

kinetic  energy  into  temperature,  but  conceals  1.00  part  poten- 
tial energy  within  the  atoms  and  molecules.  Similarly,  the 
bolometer  thread  of  metal  conceals  1.00  part  potential  energy, 
but  gives  relative  ordinates  of  kinetic  energy  in  correct  ratio 
to  the  surviving  spectrum  lines. 

The  Depletion  of  the  Solar  Radiation  from  5.5$  Calories  in  the 
Isothermal  Layers  of  the  Sun  to  1.50  Calories  at  the  Sea  Level 

of  the  Earth 

We  are  now  in  possession  of  the  data  which  are  necessary 
in  order  to  describe  with  considerable  clearness  the  course  of 
depletion  of  the  original  solar  black  body  radiation,  or  the  solar 
constant,  which  amounts  to  about  5.85  gram  calories  per  square 
centimeter  per  minute,  down  to  about  the  1.50  calories  that  are 
ordinarily  received  at  the  sea  level  of  the  earth's  atmosphere. 
This  assumes  that  the  quantities  of  the  radiation  at  the  sev- 
eral levels  in  the  sun's  and  the  earth's  atmospheres  have  been 
corrected  for  instrumental  and  observational  defects,  and  all 
reduced  to  the  scale  of  the  earth's  mean  distance  from  the  sun. 
It  will  not  be  necessary  to  recapitulate  in  this  place  the  results 
of  the  research  in  the  earth's  atmosphere,  as  these  may  be 
found  in  the  Treatise  on  Atmospheric  Circulation  and -Radiation 
1915,  and  in  Bulletins  No.  3, 1912,  No.  4,  1914,  of  the  Argentine 
Meteorological  Office.  These  are  sufficiently  summarized  in 
Tables  75,  76,  and  77,  and  Figures  23  and  24,  and  they  will  be 
briefly  explained.  In  Table  79  the  wave  lengths  in  the  spectrum 
are  taken  for  the  differences  0.05  /*,  up  to  2.50  v,  and  wherever 
the  bolometer  ordinates  of  the  four  central  columns  are  lacking 
in  Abbot's  observations  the  necessary  interpolations  have  been 
added  to  fill  out  the  columns  uniformly,  so  that  the  general 
sums  may  become  homogeneous  and  comparable.  The  columns 
under  7655°,  6950°,  5810°,  5450°,  are  computed  from  the  Wien- 
Planck  Formula,  using  the  constants  as  follows,  Ci  =  5.575  X  10~15, 
c2  =  1.4455: 

5.575  X  10" 15  5.575  X  105 


/    1.4455  \  /       6277.4 

*(ekT  - 


RECONCILIATION  OF  DATA 


219 


Since  X  =  10V,  Ci  (C.  G.  S.)  is  multiplied  by  1020  for  /*,  and 
cz  (C.  G.  S.)  is  multiplied  by  104  for  /*. 

The   column   7655°  corresponds   with   the   solar   constant, 
5.85  calories  reduced  to  its  equivalent  at  the  distance  of  the 

TABLE  79 

SUMMARY  OF  THE  DATA  OF  THE  SOLAR  RADIATION,  SHOWING  THE  DEPLETION 

FROM  THE  SOLAR  CONSTANT  AT  5.85  CALORIES  TO  1.50  CALORIES 

AT  THE  SEA  LEVEL 


Wave 
Length 

THERMODYNAMICS 

BOLOMETER 

PYRHELIOMETER 

7655° 

6950° 

Total 

Mt. 

Whitney 

Mt. 
Wilson 

Wash- 
ington 

5810° 

5450° 

0.00/u.... 
0  05 

0.000 
0.300 
0.775 
1.380 
2.998 
6.490 
9.194 
10.490 
10.594 
9.992 
9.052 
8.000 
6.969 
6.078 
5.186 
4.512 
3.838 
3.348 
2.858 
2.505 
2.152 
1.897 
1.641 
1.455 
1.268 
1.131 
0.993 
0.890 
0.786 
0.709 
0.631 
0.571 
0.511 
0.464 
0.417 
0.381 
0.344 
0.315 
0.286 
0.263 
0.240 
0.222 
0.203 
0.188 
0.172 
0.160 
0.148 
0.138 
0.127 
0.118 
0.109 

0.000 
0.100 
0.375 
0.700 
1.154 
3.018 
4.854 
6.054 
6.544 
6.499 
6.127 
5.594 
5.004 
4.442 
3.879 
3.419 
2.958 
2.604 
2.250 
1.987 
1.724 
1.529 
1.333 
1.187 
1.041 
0.932 
0.823 
0.740 
0.657 
0.594 
0.530 
0.480 
0.431 
0.393 
0.355 
0.325 
0.294 
0.270 
0.245 
0.226 
0.207 
0.191 
0.175 
0.162 
0.148 
0.158 
0.127 
0.119 
0.110 
0.103 
0.095 

0.440 
2.700 
4.350 
6.050 
6.060 
5.630 
5.050 
4.350 
3.650 
3.160 
2.670 
2.470 
2.260 
1.960 
1.660 
1.460 
1.260 
1.130 
1.030 
0.940 
0.900 
0.780 
0.710 
0.670 
0.620 
0.570 
0.530 
0.470 
0.430 
0.380 
0.350 
0.320 
0.290 
0.260 
0.240 
0.230 
0.210 
0.200 
0.180 
0.160 
0.140 
0.120 
0.100 
0.070 
0.040 

1.800 
3.410 
4.930 
5.450 
5.180 
4.720 
4.100 
3.490 
3.050 
2.600 
2.410 
2.220 
1.930 
1.630 
1.440 
1.240 
1.120 
1.020 
0.930 
0.880 
0.770 
0.700 
0.660 
0.610 
0.560 
0.510 
0.460 
0.420 
0.370 
0.340 
0.310 
0.280 
0.260 
0.230 
0.220 
0.200 
0.190 
0.170 
0.150 
0.130 
0.110 
0.090 
0.060 
0.030 

1.660 
3.150 
4.840 
5.200 
4.930 
4.500 
3.970 
3.440 
3.000 
2.570 
2.390 
2.200 
1.910 
1.620 
1.420 
1.220 
1.110 
1.010 
0.920 
0.870 
0.750 
0.690 
0.650 
0.600 
0.560 
0.510 
0.450 
0.410 
0.360 
0.330 
0.300 
0.270 
0.250 
0.230 
0.210 
0.190 
0.180 
0.160 
0.140 
0.120 
0.100 
0.080 
0.050 
0.030 

2.400 
3.870 
4.270 
4.170 
3.840 
3.450 
3.060 
2.680 
2.310 
2.160 
2.010 
1.760 
1.500 
1.380 
1.200 
1.100 
1.000 
1.000 
0.910 
0.820 
0.740 
0.680 
0.620 
0.570 
0.530 
0.480 
0.440 
0.400 
0.350 
0.320 
0.290 
0.260 
0.240 
0.220 
0.180 
0.170 
0.150 
0.130 
0.110 
0.090 
0.070 
0.040 
0.020 

0.000 
0.020 
0.050 
0.090 
0.150 
0.590 
1.319 
1.883 
2.348 
2.609 
2.685 
2.631 
2.495 
2.305 
2.114 
1.917 
1.720 
1.547 
1.374 
1.234 
1.094 
0.984 
0.873 
0.786 
0.698 
0.630 
0.562 
0.510 
0.457 
0.415 
0.373 
0.341 
0.308 
0.232 
0.256 
0.236 
0.215 
0.198 
0.180 
0.166 
0.152 
0.141 
0.130 
0.121 
0.111 
0.104 
0.096 
0.090 
0.083 
0.078 
0.072 

0.000 
0.010 
0.020 
0.040 
0.067 
0.307 
0.721 
1.179 
1.559 
1.809 
1.931 
1.947 
1.893 
1.778 
1.662 
1.526 
1.389 
1.261 
1.133 
1.024 
0.915 
0.827 
0.739 
0.668 
0.598 
0.542 
0.486 
0.442 
0.397 
0.362 
0.327 
0.299 
0.271 
0.249 
0.226 
0.208 
0.190 
0.176 
0.161 
0.149 
0.137 
0.127 
0.117 
0.109 
0.101 
0.094 
0.087 
0.081 
0.075 
0.070 
0.065 

0.10  
0  15 

0.20  
0  25 

0.30  

0.35  
0.40  

0.45  
0.50  
0.55  
0.60  
0.65  

0.70  
0.75  

0  .  80  
0.85  
0.90  
0.95  
1.00  
1.05  
1.10  . 

1.15  
1  20 

1.25  
1  30 

1.35  

1  40 

1.45  

1  .  50  
1  .  55  
1  .  60  

1.65  
1  70 

1.75  
1  .  80  
1.85  
1.90  
1.95 

2.00  
2  05 

2.10  
2  15 

2.20  
2  25 

2.30  . 

2.35  
2.40  .  . 

2.45  
2  .  50  

Sums  

123.474 

83.266 

67.250 

61.380 

59.550 

51.990 

39.773 

30.551 

220 


A   TREATISE   ON   THE    SUN  S   RADIATION 


TABLE  79 — Continued 
Factor  to  reduce  to  gr.  cal./cm.2  min.  =  20.9 


Reduced  
Thermo- 
dynamics. .  .  . 

5.91 
5.85 

3.98 
3.98 

3.22 
3.23 

2.94 
2.96 

2.85 
2.87 

2.49 

2.47 

1.90 
1.94 

1.46 
1.50 

Log  T  
Log  T*  

Log  (RV  - 

Log  \D)  T  '  ' 

Log  Jo  
Jo  

3.88395 
15.53580 

-15.23180 

0.76760 
5.856 

3.84198 
15.36792 

-15.23180 

0.59970 
3.978 

- 

- 

- 

- 

3.76418 
15.05672 

-15.23180 

0.28852 
1  943 

3.73640 
14.94560 

-15.23186 

0.17740 
1  505 

earth,  and  it  is  derived  from  thermodynamics;  the  column 
6950°  is  the  effective  radiation  on  the  outermost  stratum  of 
the  earth's  atmosphere,  and  it  is  derived  also  from  thermody- 
namics, as  indicated;  the  column  under  5810°  corresponds  with 
Abbot's  extrapolated  value  of  the  solar  constant  1.94  calories 
(compare  Abbot's  "  Sun,"  p.  298);  the  column  5450°  is  equiva- 
lent to  about  1.50  calories,  the  full  amount  measured  by  the 
pyrheliometer  on  the  sea  level.  The  four  columns  marked 
Total,  Mt.  Whitney,  Mt.  Wilson,  Washington,  are  derived 
from  Abbot's  bolometer  ordinates,  Vol.  II,  Annals  Smithsonian 
Institution,  or  the  Astrophysical  Journal,  October,  1911,  the 
method  of  reducing  the  original  arbitrary  units  to  the  probable 
calories  being  given  in  Bulletin  No.  3,  p.  83.  The  values  at 
the  several  wave  lengths  have  been  made  as  homogeneous  as 
practicable  in  Table  79  and  the  corresponding  curves  are  plotted 
on  Fig.  25,  where  their  mutual  relations  can  be  examined.  The 
sums  of  the  several  columns  are  taken,  and  as  they  are  com- 
parable there  should  be  a  common  factor  to  reduce  the  areas 
which  are  equivalent  to  the  curves  to  the  corresponding  calories. 
Such  a  factor  has  been  found  to  be  20.90.  The  factor  used 
in  Bui.  No.  4,  page  84,  is  20.72,  where  the  same  table  in  part 
may  be  found.  The  lower  portion  of  Table  79  contains  the 
reductions  in  calories  at  the  distance  of  the  earth.  Under 
"  reduced  "  values  the  sums  of  the  columns  divided  by  the 
factor  20.9  are  given;  under  "  thermodynamics  "  are  found  the 
corresponding  values,  as  deduced  from  the  several  independent 
discussions  by  thermodynamics,  from  bolometer  observations, 
and  from  pyrheliometer  reductions.  It  is  readily  seen  that  the 


RECONCILIATION   OF   DATA 


221 


Gr.  cal./cm.2  min 


cal.«=S  )lar  Constant  (Bigelow). 


3.22  cal.=0utside  atmosphere  (Abbot) 


1.94  cal.=  Solar  Constant  (Abbot) 


0.50  1.00  1.50  2.00  2.50/f 

FIG.  25.     Summary  of  the  Data  Derived  from  Thermodynamics,  Bolometers, 
and  Pyrheliometers. 


222 


A   TREATISE   ON   THE   SUN  S   RADIATION 


data  of  Table  79  constitute  a  nearly  homogeneous  scheme,  from 
the  solar  constant  in  the  isothermal  layers  of  the  sun  to  the 
bottom  of  the  earth's  atmosphere  at  the  sea  level. 

The  reduced  values  of  Table  79  are  to  be  compared  with 
the  computed  values  derived  from  thermodynamics  and  the 
observations  with  bolometers  and  pyrheliometers. 

The  Line  and  Band  Absorption  and  the  Scattering 

In  addition  to  the  amount  of  the  scattering  and  the  absorp- 
tion already  indicated,  there  is  a  small  additional  amount  re- 
corded as  depleted  areas  of  a  triangular  shape  within  the  smooth 
spectrum  curves  heretofore  assumed.  Scattering  and  absorp- 
tion modify  the  original  effective  solar  spectrum,  3.98  calories 
on  the  earth's  vanishing  plane,  by  depressing  many  ordinates 
more  or  less,  and  by  selective  depletion  at  certain  lines  and 
bands.  Mr.  F.  E.  Fowle  has  studied  these  spectrum  regions 
with  the  following  results.  The  lines  and  bands,  together  with 
their  absorbent,  oxygen  or  aqueous  vapor,  are  here  given. 

THE  LINE  AND  BAND  ABSORBENTS 


Line 

n 

Absorbent 

Line 

M 

Absorbent 

B  ? 

0.69 

Oxygen 

$  

1.13 

Water  Vapor 

a 

0  72 

Water  Vapor 

i/'..  . 

1.42 

Water  Vapor 

A 

0  76 

Oxygen 

J2  

1.89 

Water  Vapor 

0.81 

Water  Vapor 

OJi  

2.01 

1 

0  93 

Water  Vapor 

ti>2  

2.05 

•? 

Astrophysical  Journal,  December,  1915. 

By  referring  to  Fig.  25  the  position  of  these  lines  can  be 
located,  and  their  relative  ordinates  for  the  stations,  Mt.  Whit- 
ney, Mt.  Wilson,  and  Washington,  can  be  calculated  from  Table 
79,  the  factor  of  reduction  to  calories  being  20.9.  Mr.  Fowle 
gives  the  percentages  under  certain  conditions,  and  the  corre- 
sponding calories  referred  to  1.930  calories  for  the  temperature 
spectrum  corresponding  with  5800°.  It  is  believed  that  this  ref- 
erence curve  is  unconfirmed  in  thermodynamics,  and  while  adopt- 
ing Mr.  Fowle's  relative  ordinates  as  percentages,  we  reduce  them 


RECONCILIATION  OF  DATA 


223 


to  the  ordinates  of  Fig.  25,  at  least  approximately,  according 
to  the  bolometer  ordinates  of  the  station,  Mt.  Whitney  2.96 
calories,  Mt.  Wilson  2.87,  Washington  2.47,  as  given  in  Table  80. 

TABLE  80 

MEAN  DEPLETIONS  DUE  TO  SCATTERING  AND  ABSORPTIONS  IN  PERCENTAGES 

AND  IN  CALORIES 

I.     Reduced  to  Calories  Observed  by  the  Bolometer 


Station 

Precipi- 
table 
Water 
in  Cm. 

DRY  AIR 

AQUEOUS  VAPOR 

Scatt. 

Absd. 

Scatt. 

Absd. 

Mount 
Whitney 
4420  meters 

0.00 
0.11 
0.25 
0.50 

7.3% 

0.5% 

-% 
0.5 
0.5 
1.0 

-% 
4.1 
5.2 
6.2 

7.3 

0.5 

0.7 

5.2 

Mount 
Wilson 
1720  meters 

0.00 
0.33 
0.50 
1.00 
2.00 

7.8 

0.5 

1.0 
1.6 
2.1 
4.7 

5.7 
6.2 

7.8 
9.3 



7.8 

0.5 

2.4 

7.3 

Washington 
Sea  Level 

0.00 
0.50 
1.80 
2.40 

9.3 

0.5 

4.1 
13.5 
19.7 

6.2 

7.8 
8.3 

9.3 

0.5 

12.4 

7.4 

Reduced  to 
Calories 

gr.  cal. 

DRY  AIR 

AQUEOUS  VAPOR 

TOTAL 

cm.2  min. 

Scatt. 

Absd. 

Scatt. 

Absd. 

Scatt.         Absd. 

Mt.  Whitney  
Mt.  Wilson  
Washington  

2.96 

2.87 
2.47 

0.216 
0.224 
0.230 

0.015 
0.014 
0.012 

0.021 
0.069 
0.306 

0.154 
0.209 
0.182 

0.237      0.169 
0.293      0.223 
0.536      0.194 

II.     Reduced  to  the  Calories  Observed  by  the  Pyrheliometer 


Mt.  Whitney  
Mt.  Wilson  

1.72 
1.64 

0.126 
0.128 

0.009 
0.008 

0.012 
0.039 

0.089 
0.120 

0.138 
0.167 

0.098 
0.128 

Washington  

1.50 

0.140 

0.007 

0.186 

0.111 

0.326 

0.118 

224 

In  Section  I  the  percentages  of  the  scattering  and  the  ab- 
sorption, as  measured  from  the  bolometer  ordinates,  have  been 
reduced  to  gr.  cal./cm.2  min.;  in  Section  II  the  same  percent- 
ages have  been  applied  to  the  mean  values  of  the  radiation 
intensity  observed  by  the  pyrheliometer.  These  mean  values 
give  a  general  idea  of  the  values  to  be  expected,  and  they  must 
be  applied  to  the  observations  by  the  bolometer  and  the  pyr- 
heliometer, respectively,  in  such  a  way  that  both  series  shall 
give  the  same  value  of  the  effective  solar  radiation,  which  should 
be  alike  in  all  strata  from  the  sea  level  to  the  vanishing  plane 
of  the  earth's  atmosphere. 

We  may  now  confirm  the  foregoing  analysis  of  the  thermo- 
dynamic  data  and  the  observed  data  of  radiation  as  given,  by 
comparing  these  with  the  data  of  the  bolometer  as  contained 
in  Table  79  and  Figs.  25,  26.  The  several  regions  from  I  to 
VII  have  been  marked  on  them  respectively. 

I.  Kinetic   energy  up  to  the  curve  5450°,  equal  to  1.50 
calories. 

II.  Potential  energy  between    the   curve   5450°  and   the 
bolometer  curve  for  Washington,   2.47  —  1.50  =  0.97  calorie 
as  compared  with  0.95  calorie  on  Table  79. 

III.  The    specific    heat    term,    (Cp  —  Cv)  =  R,    which    is 
made  up  from  the  absorption  of  the  short  wave  ordinates.  This 
amounts  on  the  sea  level  to  1.19  —  0.19  =  1.00  calorie.     The 
summation  of  the  short  wave  ordinates  from  0.00  /*  to  0.38  /z 
is  about  20.231;  if  this  is  divided  by  the  factor  20.9  the  result 
is  0.98  calorie.     The  absorption  of  the  short  wave  ordinates  is 
necessary  in  order  to  change  the  specific  heats  of  the  atmosphere 
from  that  of  the  inner  energy  Cv,  Uu  =  Cv  p  T,  into  that  of 
the  gravitation  energy  Up,  —  g  (zi  —  z0)  =  Cp  (Ta  —  T0). 

IV.  The  reflected  or  scattered  energy. 

V.  The   absorbed    energy  used  in  molecular  and  atomic 
forms.     The  detailed  distribution  of  these  energies  in  the  lowest 
strata  will  require  more  study.     On  the  Washington  level  U  = 
H  +  /  +  s  =  2.62  and  on  Table  79,  2.49;    on  the  Mt.  Wilson 
level    U  =  2.82  and  on  Table  79,  2.85;    on  the  Mt.  Whitney 
level  U  =  2.98  and  on  Table  79,  2.94.     These  points  are  indi- 


RECONCILIATION  OF  DATA 


225 


cated  on  Fig.  26.     Compare  the  results  from  thermodynamics. 

VI.     The  transition  region  from  the  electromagnetic  energy 

to  the  thermodynamic  energy  of  ionization  may  provisionally 


70000 


60000 


50000 


40000 


30000 


20000 


10000 


FIG.  26.     The  Seven  Components  Which  Make  up  the  Intensity  of  the  Solar 
Radiation  at  the  Earth,  3.98  gr.  cal./cm.2  min. 

be  assigned  to  the  high-level  absorptions  along  the  slope  of  the 
ordinates  from  the  wave  length  0.30  /*  to  0.50  /*. 

VII.  The  great  depletion  from  the  total  solar  intensity, 
5.85  calories,  to  the  effective  intensity  at  the  earth,  3.98  calories, 
making  the  difference  1.87  calories,  is  contained  in  the  area 
between  the  7655°  and  the  6950°  black  body  curves. 

Direct  Readings  of  the  Pyrheliometers  at  Great  Heights 

In  a  paper,  "  New  Evidence  on  the  Intensity  of  Solar  Radia- 
tion Outside  the  Atmosphere,"  1^15,  Mr.  Abbot  summarizes 
the  results  of  his  direct  readings  of  pyrheliometers  on  balloons 


226 


A  TREATISE  ON  THE  SUN'S  RADIATION 


up  to  about  22000  meters.  Taking  these  in  connection  with 
the  pyrheliometer  readings  in  the  zenith  at  Mt.  Whitney,  Mt. 
Wilson,  Washington,  D.  C.,  we  have  the  following  data: 

TABLE  81 

SUMMARY  OF  THE  DIRECT  READINGS  OF  THE  PYRHELIOMETERS  WHEN  REDUCED 
TO  THE  ZENITH,  AND  WITHOUT  ANY  EXTRAPOLATION  TO  THE  SUPPOSED 
SOLAR  CONSTANT. 


Location  of  the  Pyrheliometer 

Height  in 
Meters 

Zenith  reading 
gr.  cal./cm.2  min. 

Abbot  balloon  ascension,  July  11,  1914  

22000 

.84 

Peppier  balloon  ascension,  Oct.  19,  1913.  . 

7500 

.76 

Mt.  Whitney,  maximum 

4420 

72 

Mt.  Wilson,  maximum 

1780 

64  (modified) 

Washington,  D.C.,  or  sea  level   . 

0 

.58  (modified) 

The  data  derived  by  the  self-registering  pyrheliometer  car- 
ried on  balloons  serve  to  fix  new  points  along  the  graph  that 
has  heretofore  been  assigned  to  the  pyrheliometer  curves,  as 
in  Fig.  26,  so  that  the  main  features  of  the  problem  have  not  been 
in  the  least  modified  by  their  addition.  The  problem  is  the  recon- 
ciliation of  the  pyrheliometer  and  the  bolometer  observations 
without  making  the  assumption  that  the  sun  does  not  radiate 
energy  as  a  black  body.  The  mean  value  of  I  at  Mt.  Wilson 
is  1.53,  the  maximum  being  1.64,  and  the  mean  value  at  Wash- 
ington is  1.33,  the  maximum  being  about  1.58  calories. 

Summary  of  the  Pyrheliometer  Results  as  Reduced  by 
Bigelow's  Method 

The  result  of  the  discussion  of  the  pyrheliometer  observations 
at  different  stations,  by  Bigelow's  method  of  reduction,  is  col- 
lected in  Table  37,  Bulletin  No.  4,  0.  M.  A.,  of  which  the 
following  is  an  extract. 

The  height  z  is  in  meters,  and  the  free  heat  received  by  the 
pyrheliometer  is  70,  after  the  depletions,  due  to  aqueous  vapor, 
dust,  and  other  matter  near  the  station,  have  been  eliminated. 
It  may  be  noted  that  by  the  Bigelow  method  of  reduction  the 
pyrheliometer  is  made  to  record  the  kinetic  energy  of  region 


RECONCILIATION   OF   DATA 


227 


TABLE  82 

SUMMARY  OF  THE  AMOUNTS  OF  THE  SOLAR  RADIATION  AS  DETERMINED  BY 
PYRHELIOMETER  OBSERVATIONS 


Station 

z 

la 

La  Confianza 

4483 

2.142 

gr.  cal 

Mt.  Whitney 

4420 

2.138 

cm.2  min. 

La  Quiaca  
Humahuaca  . 

3465 
2939 

2.005 
1.930 

Maimara  

2384 

1.852 

These  are  plotted  on  Fig 

Mt.  Wilson  

1780 

1.771 

26  as  crosses  x  x  x  x  x 

Tuiuv 

1302 

1.705 

while  the  Abbot  data  ap- 

Bassour   
Mt.  Weather 

1160 
526 

1.675 
1  601 

pear  on  the  graph  which 
is  marked  "  Pyrheliom- 

Cordoba  

438 

1.592 

eter." 

Filar  

340 

1.572 

Potsdam 

89 

1.538 

Washington  

34 

1.529 

Sea  level 

o 

1  525 

I  and  in  part  the  potential  energy  of  region  II.  It  is  apparent 
that  the  course  of  the  line  of  crosses  on  Fig.  26  is  seeking  the 
line  (H  +  /)  which  bounds  the  potential  region.  It  is  to  be 
hoped  that  suitable  observations  can  be  made  in  balloon  as- 
censions, which  will  enable  this  idea  to  be  verified.  Abbot's 
observation  is  along  the  line  H,  while  Bigelow's  reduction  has 
a  very  different  vertical  gradient  from  Abbot's,  that  is,  along 
the  dotted  line. 

Computation  of  the  Pyrheliometer  Data,  Leading  to  3.98  gr.  cal./ 
cm.2  min.  as  the  Intensity  of  the  Solar  Radiation  at  the  Earth 

The  results  of  our  computation  are  contained  in  Table  83 
and  Fig.  26.  The  data  of  Tables  60-69,  Bulletin  No.  4,  Oficina 
Meteorologica  Argentina,  have  been  adopted,  except  for  the 
stations  Cordoba  and  La  Quiaca,  where  the  individual  obser- 
vations were  computed  throughout  the  years  1912,  1913,  1914, 
and  1915,  and  the  general  means  of  the  entire  series  are  placed 
in  Table  83.  This  table,  therefore,  represents  the  average  con- 
dition at  each  station  after  the  local  and  periodic  action  has 
been  eliminated.  The  method  applied  to  the  daily  observations 
is  equally  valid. 


228 


A   TREATISE   ON  THE   SUN  S   RADIATION 


We  have  exhibited  the  following  components,  into  which 
the  solar  radiation  divides  itself  in  the  earth's  atmosphere: 

1.  H  =  the  kinetic  energy  of  the  translation  of  the  mole- 
cules, measured  by  the  pyrheliometer  as  temperature.  It  is 
evident  that  this  instrument  is  a  means  of  registering  only  kinetic 
energy,  and  that  the  various  forms  of  the  potential  energy  in 
the  atmosphere  do  not  come  into  its  competency.  The  bol- 
ometer, on  the  other  hand,  by  means  of  its  relative  ordinates, 
does  indicate  some  portions  of  this  potential  energy  in  its  results, 
though  in  a  confused  fashion.  The  thermodynamics  has  the 
power  to  classify  clearly  the  distribution  of  the  various  forms 
of  the  potential  energy  in  all  the  strata  of  the  atmosphere. 


TABLE  83 

COMPUTATION  OF  THE  PYRHELIOMETER  DATA  TO  INCLUDE  THE  POTENTIAL 
ENERGY,  THE  SPECIFIC  HEAT,  AND  THE  IONIZATION 


2 

* 

e 

k 

H 

J 

H+J 

5 

a 

.V, 

R 

5 

(a+R)' 

Name 

i 

"*"*     W) 

gl 

of  the 

.  1 

0) 

*O 

& 

|i 

3 

"S 

la 

.0 

•2  . 

*J  « 

Station 

.fl  ^ 

3 

£ 

'£    M 

C3    ^2 

gJO 

<B  §2 

»-i    4J 

tfi 

15 

&& 

& 

3 

a 

o  2 

Uc/5 

II 

II 

II 

II 

II 

II 

C/3  ffi 

11 

11 

La  Confianza  .  .  . 

4485 

-20° 

1.7 

.072 

1.741 

1.116 

2.857 

.205 

.080 

-.004 

0.840 

3.988 

0.920 

Mt.  Whitney   .  . 

4420 

+37 

2.1 

.088 

.668 

1.069 

2.737 

.241 

.120 

+  .001 

0.880 

3.979 

1.000 

La  Quiaca  

3462 

-22 

3.3 

.103 

.662 

1.065 

2.727 

.281 

.080 

-.008 

0.900 

3.980 

0.980 

Mt.  Wilson..   .. 

1727 

+34 

5.8 

.119 

.534 

.983 

2.517 

.300 

.125 

+  .010 

1.020 

3.972 

1.145 

Bassour  

1160 

+36 

7.8 

.172 

.386 

.889 

2.275 

.392 

.155 

+  .010 

1.145 

3.977 

1.300 

Mt.  Weather  .  . 

526 

+39 

8.5 

.178 

.322 

.847 

2.169 

.386 

.255 

+  .006 

1.175 

3.991 

1.430 

Cordoba  

438 

-31 

9.4 

.161 

.446 

.927 

2.373 

.382 

.102 

-.005 

1.125 

3.977 

1.230 

Washington.    .  . 

34 

+39 

4.7 

.201 

.329 

.852 

2.181 

.438 

.177 

-.004 

1.188 

3.980 

1.365 

z  =  the  height  in  meters;  <f>  =  the  latitude  of  the  station. 

e  =  the  vapor  pressure  in  millimeters.  Unless  this  is  reduced  to  a  mean  value  for  the  station 
there  will  be  an  annual  variation.  The  successive  mean  eo  are  2,  2,  4,  5,  7,  8,  10,  5  mm.  for 
these  stations.  The  coefficient  0 . 012  was  determined  as  the  equivalent  in  calories  for  1  mm. 
of  e.  Hence  A«  =  0.012  (e-e0). 

k  =  the  coefficient  of  scattering,     s  =  k  (H  + J). 

H  =  7i,  the  kinetic  energy.  This  is  the  value  I\  of  the  zenith  intensity  of  the  radiation, 
observed  by  the  pyrheliometer,  and  reduced  by  the  Bouguer  Formula  /  =  /o  p8ecz .  No  extra- 
polation is  made  beyond  sec  2  =  1. 

J  =  the  potential  energy,  J  =  0.641  H. 

H  +  J  =  U  the  total  inner  energy  after  its  losses  by  scattering  and  absorption. 

(a  +R)  =  the  specific  heat  effect  plus  the  absorption.  R  is  computed  from  thermodynamic 
data  as  in  Table  75,  and  elsewhere,  and  an  approximate  value  has  been  assigned  for  each 
station.  •  If  the  mean  solar  constant  at  the  earth  is  3.980  calories,  then  we  have 
a  =  S-(H+J+s  +  Ae+R).  The  absorption  in  Table  83  has  been  obtained  in  this  manner 
and  it  is  in  harmony  with  Fowle's  results. 


RECONCILIATION  OF  DATA  229 

2.  /  =  the  potential  energy  in  the  atmosphere,  correspond- 
ing with  the  kinetic  energy  through  the  formula,  /  =  0.641  H. 

3.  s  =  the  scattering,  or  reflection  of  the  radiation  on  the 
air  contents,  without  affecting  the  temperature  as  such  on  the 
levels  below  that  of  the  reflection.     If  the  coefficient  is  k  as 
obtained  from  k  =  (1.00  —  p),  where  p  is  computed  from  the 
pyrheliometer  observations,  then  s  —  k  (H  +  J)  the  part  which 
is  not  transmitted. 

a  =  the  true  absorption,  as  shown  by  the  band  and  line 
depletions  of  the  ordinates  of  the  several  stations.  It  is  com- 
puted indirectly  from  the  adopted  mean  intensity  of  the  solar 
radiation, 

S  =  3.980  calories,  by  the  formula, 

(167)  a  =  S  -  (H  +  J  +  k  (H  +  f)  +  A  e  +  R). 

A  e  =  the  vapor  pressure  correction  to  a  mean  value  of  £<>, 
with  a  coefficient  0.012, 

(168)  Ae  =  0.012  (e  -  e0). 
R  —  the  specific  heat  energy, 

(169)  Ru  (Ta  -  To)  =  (Cpa  -  Cv1Q)  (Ta  -  To), 

which  is  necessary  to  change  the  inner  energy  at  the  specific 
heat  of  constant  volume  Uv  into  the  inner  energy  at  the  specific 
heat  of  constant  pressure  UP.  The  Boyle- Gay  Lussac  Law 
in  the  free  gas  leads  to  Uv,  where  there  is  no  gravitation,  but 
to  Up  where  there  is  gravitation,  as  by  formulas,  (l)  to  (10). 
Thus,  we  have,  generally, 

(170)  Uv  =  Cv.PT=  (Cp  -R}PT 

(171)  Up  =  CpPT=  (Cv  +  R)PT  =  -gpz 

It  is  Uv  which  is  depleted  by  scattering,  absorption,  and 
ionization,  so  that  we  have, 

(172)  Uv  =  H  +  J  +  s  +  a  +  &e  +  E  (ionization.) 
To  obtain  the  stratum-intensity  of  the  radiation. 

(173)  S=UV  +  RPT, 

in  conformity  with  final  equilibrium  against  the  impressed  force 
of  gravitation. 

It  should  be  noted  that  we  have  arrived  at  the  solar  radia- 
tion S,  and  by  Table  83  find  it  to  be  3.980  calories  on  each  stratum. 
In  this  process  there  has  been  no  extrapolation  by  the  Bouguer 


230  A  TREATISE   ON  THE   SUN'S   RADIATION 

formula  beyond  its  legitimate  mathematical  function  in  the  zenith, 
so  that  all  difficulties  on  that  score  are  fully  obviated. 

E  =  the  ionization,  in  the  great  region  marked  VI,  which 
lies  between  (H  +  7)  +  (s  +  a)  and  Uv. 

There  are  two  points  to  be  noted  in  respect  of  the  ionization: 
(1)  Since  the  potential  energy  of  the  electromagnetic  radiation 
is  1.00  on  a  scale  of  2.00,  and  the  atmospheric  potential  energy 
is  0.641,  it  follows  that  there  is  a  change  of  0.359  on  the  same 
scale.  Hence,  we  have, 

E  =  0.359  X^    X  3.980  =  0.714  -gr'  °a  ' 


the  amount  of  the  solar  radiation  which  is  probably  transformed 
into  electric  energy  of  the  free  ions  in  the  upper  levels,  as  pro- 
duced by  the  action  of  the  short  waves  upon  the  atmospheric 
atoms  and  molecules. 

(2)  In  Fig.  26,  the  transition  between  the  electromagnetic 
potential  energy  and  the  thermodynamic  potential  energy  is 
represented  as  abrupt  or  discontinuous.  This  is  probably  not 
the  case  in  nature,  and  it  is  likely  that  a  gradual  change  occurs 
in  the  coronal  region,  between  90000  and  50000  meters,  while 
the  process  is  intensified  along  the  upper  branch  of  the  isothermal 
layer,  50000  to  37000  meters,  where  the  temperature*  is  rapidly 
changing  its  value.  //  the  production  of  ions  is  to  be  associated 
•with  rapid  changes  of  temperature  in  atmospheres,  we  have  five 
conspicuous  regions  where  ionization  occurs: 

1.  At  the  bottom  of  the  solar  isothermal  layer. 

2.  At  the  top        "    " 

3.  At  the  top        "  the  terrestrial  isothermal  layer. 

4.  At  the  bottom  "    " 

5.  At  the  bottom  of  the  atmosphere  and  in  the  earth.    These 
are  the  regions  where  electric  charges  are  known  to  accumulate 
and  to  manifest  themselves,  as  solar  static  electric  fields,  as  cur- 
rents -of  electricity  inducing  variations  in  the  normal  magnetic 
field,  as  auroras  in  the  high  level  electricity  in  the  earth's  at- 
mosphere, thunderstorm  energy  in  the  convectional  region,  and 
at  the  surface  of  the  earth  itself. 


RECONCILIATION  OF  DATA  231 

Synchronism  of  the  Solar  and  the  Terrestrial  Variations  during 
the  Interval  1900-1915  in  Argentina 

It  is  of  interest  to  extend  the  computation  of  the  intensity 
of  the  radiation  to  the  annual  values,  in  order  to  compare  the 
variations  with  the  other  series  of  annual  variations  derived 
from  various  solar  and  terrestrial  sets  of  data.  This  can  be 
done  immediately  for  the  years  1903-1916,  and  the  record  can 
be  carried  back  to  1883  by  means  of  the  observations  at  Mont- 
pellier,  Geneva,  and  Warsaw.  Table  84  contains  the  summary 
of  the  annual  values.  It  may  be  recalled  that  the  computa- 
tion here  employed  is  the  third  attempt  to  make  the  pyrheli- 
ometer  data  conform  to  the  requirements  of  solar  and  terrestrial 
thermodynamics,  which  sustain  the  bolometer  data  in  requir- 
ing about  4.00  calories  for  the  effective  solar  radiation  at  the 
earth,  in  place  of  the  2.00  calories  derived  from  the  pyrheli- 
ometer  as  interpreted  by  the  Langley-Abbot  method  of  dis- 
cussing the  Bouguer  formula  of  depletion.  The  first  method 
introduced  a  functional  correction  to  70  due  to  the  vapor  pressure 
effect, 

(174)  /  =  (Ic  -  0.0214  e)  ^sec2,  (I0  =  Ic  -  0.0214  e). 

The  second  method  was  an  attempt  to  reconcile  the  very  dis- 
cordant results  derived  from  stations  at  different  altitudes,  and 
consisted  in  making  pw  a  function  of  the  vapor  pressure, 

(175)  7  =  /0  (#.-/«)"- 

where  each  station  has  an  equation  for/  (e),  Bulletin  No.  4,  0.  M. 
A.,  Chapter  5;  Treatise,  Chapter  VII.  Both  of  these  are  methods 
of  extrapolation,  but  the  third  method,  described  above,  com- 
pletely rejects  every  attempt  to  extrapolate  beyond  sec  z  =  1, 
and  it  succeeds  in  reducing  each  station  on  its.  own  level  to  the  solar 
intensity  independently  of  other  stations,  and  without  making  any 
use  of  certain  bolometer  factors,  which  are  excessively  complex,  and 
impractical  of  attainment  under  ordinary  conditions  of  pyr- 
heliometer  observations. 

In  transferring  these  results  to  Fig.  27,  the  value  3.914  for 
Mt.  Whitney,  1910,  seems  to  be  rather  too  small,  due  to  the 


232 


A   TREATISE   ON  THE   SUN'S   RADIATION 


TABLE  84 
MEAN  ANNUAL  VALUES  AS  COM 


001>CO 
l>  1—  1  1—  1 
O5OO 


CO  ^^  CO  CO  CO 


O5         O5 
CO        CO 


§  i 


CO        CO        CO        CO 


CO        CO        CO 


CO 


j 


li 

flj       ^3 

S-S 


&rC 

ri 

§"S 

-M  •£ 

3  § 

T3  T3 
C 

^'2 

M       IH 

tj    rt 

0)      CO 

'Si 
^  S 

«i 

cti 

I"8 

5   rt 

1-s 
fi- 
ll 

w    en 

gr§ 

tx§ 

£  '5 

QJ      ^ 

•4-J       3 

rt    cx 

2  i 

3   fe 

rt    _T! 


.2    S 

•g  § 
«s 

15 
II 


fl! 


RECONCILIATION  OF  DATA  233 

limited  number  of  the  observations,  but  it  is  retained;  the 
value  for  Mt.  Weather,  1912,  3.790,  seems  to  have  been  unduly 
depressed  by  the  dust  from  the  Volcano  Katmai.  There  are 
local  conditions  of  this  sort  which  tend  to  distort  the  annual 
means,  and  as  they  cannot  be  eliminated  from  the  normal  in 
any  simple  manner,  it  becomes  necessary  to  utilize  a  larger 
number  of  stations  for  pyrheliometer  records  in  various  parts 
of  the  world.  These  should  be  made  and  reduced  on  an  ac- 
cepted uniform  plan,  which  conforms  to  the  requirements  of 
thermodynamics  as  well  as  of  homogeneous  statistics.  The 
annual  values  in  Table  84  produce  a  curve  which  is  similar 
in  its  general  features  to  the  one  in  Table  93  of  the  Treatise, 
though  it  here  presents  an  additional  minor  maximum  for  the 
year  1911. 

We  shall  next  proceed  to  compare  these  results  with  other 
easily  accessible  data,  as  given  in  Table  85  and  Fig.  27.  Similar 
tables  of  comparison  have  been  prepared  for  Argentina,  from 
1870  to  1916,  and  they  present  quite  the  same  general  facts 
of  synchronism  throughout  this  interval. 

It  is  necessary  to  receive  prompt  reports  of  the  frequencies 
of  the  sun-spots  and  the  prominences  in  order  to  utilize  them 
in  any  forecast  service  for  the  use  of  the  public. 

There  are  several  facts  which  appear  quite  distinctly. 

(1)  The  sun-spot  frequencies,  the  prominence  numbers,  the 
solar  radiation,  the  amplitudes  of  the  horizontal  magnetic  force, 
the  precipitation,  temperature,  vapor  pressure,  and  barometric 
pressure  all  agree  in  conforming  to  the  long  period  of  11  years. 

(2)  They   all   conform   with   certain   local   irregularities  in 
respect  of  maxima  and  minima  in  the  subordinate  3.75-year 
period. 

(3)  The  sun-spots,  prominences,  magnetic  field,  and  baro- 
metric pressure  all  move  together  in  a  direct  synchronism,  while 
the  radiation,  the  precipitation,  the  temperature  in  part,  and 
the  vapor  pressure  move  in  an  inverse  synchronism. 

(4)  Taking  the  entire  set  of  curves  as  attempting  to  express 
a  fundamental  law  of  solar  activity,  it  is  thought  that  a  minor 
crest  occurs  at  the  maximum  of  the  11 -year  period,  that  a 


234 


A   TREATISE   ON  THE   SUN'S  RADIATION 


TABLE  85 

SUMMARY  OF  THE  SOLAR  AND  TERRESTRIAL  OBSERVATIONS  SHOWING  A 

SYNCHRONISM  IN  THE  VARIATIONS  IN  THE  H-YEAR  AND  THE 

3.75-YEAR  PERIODS 


Year 
of 
Observation 

Sun- 
spots 

Promi- 
nences 

Radia- 
tion 

Hori- 
zontal 
Mag. 
Ampl. 

Precipi- 
tation 

Temper- 
ature 

Vapor 
Pressure 

Baro- 
metric 
Pressure 

1900.. 

114 

128 

1589 

855 

17  81 

10  50 

749  28 

1901  

33 

88 

1374 

573 

17  87 

9  88 

49  43 

1902 

60 

47 

1464 

716 

17  68 

10  35 

48  59 

1903 

293 

118 

3  989 

1926 

775 

17  49 

10  52 

49  22 

1904  
1905  

503 

762 

291 
302 

3.979 
3  955 

1787 
2194 

914 
719 

17.17 
16  75 

10.30 
10  22 

49.12 
49  24 

1906 

646 

261 

3  975 

2015 

613 

17  55 

10  07 

49  33 

1907  
1908  

745 
583 

427 
346 

3.971 
3  940 

2299 
2584 

648 
615 

16.83 
17  11 

9.86 
9  55 

49.52 
49  97 

1909  

527 

365 

3  960 

2334 

566 

17  08 

9  34 

49  90 

1910 

223 

259 

3  952 

2354 

651 

16  96 

9  31 

49  59 

1911.. 

68 

163 

3  960 

1923 

751 

16  40 

9  26 

49  75 

1912  

43 

126 

3  978 

1520 

811 

17  51 

10  09 

49  72 

1913  

17 

135 

4  003 

1564 

704 

18  19 

10  53 

49  44 

1914  . 

115 

198 

4  002 

1672 

928- 

17  28 

10  78 

48  77 

1915  

350 

250 

3  990 

2635 

773 

17  32 

10  06 

48  92 

1916  

540 

400 

3.920 

2676 

486 

17.65 

8.87 

49.00 

Sun-spots. — Data  from  Wolfer's  Frequency  Numbers,  Meteorologische  Zeitschrift,  May, 
1902,  May,  1915. 

Prominences. — Data  from  Ricco's  Distribuzione  delle  protuberanze,  Mem.  della  Soc.  degli 
Spettro.  Italiani,  March,  1914. 

Radiation. — Bigelow's  reduction  of  the  pyrheliometer  data. 

Horizontal  Magnetic  Amplitudes. — Bigelow's  compilation  from  European  observatories, 
and  Pilar,  Argentina. 

Precipitation,  Temperature,  Vapor  Pressure,  and  Barometric  Pressure,  from  the  Argentina 
stations,  Tucuman,  Andalgala,  Goya,  Concordia,  Cordoba,  Pilar,  Buenos  Aires,  Victorica, 
Bahia  Blanca,  Patagones. 

second  crest  occurs  near  the  top  of  the  ascending  branch,  that 
a  third  crest  occurs  near  the  top  of  the  descending  branch,  and 
a  fourth  crest  near  the  bottom  of  the  descending  branch.  Some- 
times one  or  the  other  of  these  minor  crests  fails  in  the  record, 
or  it  is  distorted  by  the  local  conditions,  or  the  sun  itself  does 
not  always  xwork  upon  the  normal  plan,  so  that  the  result  is 
complex.  Each  Argentine  station  clearly  resembles  this  model 
under  the  influence  of  minor  distortions,  and  they  support  this 
proposition. 

(5)  In  the  case  of  the  temperature,  it  is  debatable  whether 
the  minor  crests  are  not  inverse  on  the  long  11 -year  period,  so 


RECONCILIATION    OF   DATA 


235 


1900 


1905 


1910 


1915 


1920 


FIG.  27.     Synchronism  of  the  Solar  and  Terrestrial  Variations. 


236  A   TREATISE   ON  THE   SUN.'s   RADIATION 

as  to  become  practically  direct.  This  must  be  due  to  the  in- 
fluence of  precipitation  and  clouds,  making  the  temperature 
at  the  surface  beneath  them  opposite  to  similar  temperature 
data  above  the  cloud  sheet,  if  it  could  be  systematically  recorded. 
(6)  If  the  rule  here  proposed  conforms  to  natural  solar  and 
terrestrial  variations,  it  follows  that  one  can  readily  construct 
a  new  solar  curve  in  advance  of  the  current  year,  giving  it  the 
mean  11-year  and  the  mean  3.75-year  periods,  so  that  a  long- 
range  forecast  of  general  annual  conditions  becomes  practical. 
In  this  way  the  11-year  maximum  is  in  1918  and  the  four  minor 
maxima  occur  in  1916,  1918-19,  1920-21,  and  1923.  These 
should  be  the  years  of  minimum  rainfall  in  Argentina,  relative 
temperature  maxima,  low  vapor  pressure  and  high  barometric 
pressure.  //  sufficient  observations  are  maintained  to  keep  in 
touch  with  any  eccentric  behavior  on  the  part  of  the  sun,  they  will 
enable  us  to  modify  this  general  statement  from  year  to  year,  and 
it  is  generally  true  that  a  forecast  for  a  year  or  two  is  without  doubt 
possible.  Local  peculiarities  will  require  a  special  study.  The 
effect  of  the  ocean  upon  coast  cities,  and  the  effect  of  mountain 
ranges  upon  cities  located  near  them,  make  such  stations  less 
valuable,  as  they  tend  to  disturb  the  smoothness  of  the  result. 
This  is  easily  seen  at  Buenos  Aires  and  Bahia  Blanca  on  the 
one  hand,  and  at  Mendoza  and  San  Juan,  near  the  Cordilleras, 
on  the  other  hand.  The  inland  stations  are  less  liable  to  dis- 
turbances, and  these  respond  to  the  solar  impulses  with  greater 
regularity.  It  is  noted  that  the  long  period  comes  out  much 
better  in  Patagonia  than  in  northern  Argentina,  in  the  case  of 
the  barometric  pressure;  while  the  latitude  of  Buenos  Aires  is 
more  favorable  for  the  periods  of  the  vapor  pressure.  It  is 
believed  that  with  suitable  experience  such  a  system  of  long- 
range  forecasts  is  easily  available  for  Argentina  and  many 
other  countries. 

Short-  and  Long-Range  Forecasts 

The  practical  application  of  solar  physics  in  meteorology 
consists  in  developing  a  system  of  interpretation  of  solar  phe- 
nomena, such  that  the  variations  of  the  solar  activity  may  be- 


RECONCILIATION  OF  DATA  237 

come  intelligible  in  terms  of  climatic  effects.  The  complexity 
of  the  physical  mechanism  is  so  great  that  the  progress  toward 
the  solution  is  slow,  although  there  is  no  reasonable  doubt  that 
an  effective  synchronism  exists.  The  efforts  to  justify  this 
procedure  have  been  entirely  statistical,  whereby  a  succession 
of  maxima  and  minima  has  been  determined  in  a  general  way, 
coordinating  the  two  parts  of  the  process.  Many  irregularities 
are  encountered,  and  in  a  country  of  violent  cyclonic  aption, 
like  the  United  States,  the  symptoms  are  much  confused^  in 
Europe  they  are  nearly  extinct,  while  in  Argentina  they  are 
comparatively  simple  and  distinct.  Criticism  is  so  mixed  up 
with  an  incomplete  knowledge  of  the  science  that  the  growth 
of  the  research  is  unduly  retarded;  the  enormous  accumulation 
of  the  statistical  data  makes  its  treatment  very  oppressive; 
the  data  are  not  sufficiently  homogeneous  to  bring  out  clearly 
trie  small  true  solar  residuals;  the  observations  are  made  in  the 
interests  of  short-range  forecasts  to  the  entire  neglect  of  the 
long-range  forecasts  of  solar  physics;  the  several  branches  of 
the  subject  are  so  scattered  as  to  be  unmanageable,  the  solar 
physics,  the  bolometric  spectra,  the  magnetic  field,  the  electric 
ionization,  the  climatic  meteorology,  all  being  separated  among 
different  administrative  offices ;  the  published  reports  from  these 
different  sources  are  so  retarded  that  they  are  useless  in  any 
study  of  current  forecasts  whatsoever;  there  exists  no  systematic 
machinery  for  handling  this  kind  of  world-wide  material. 

It  is  inevitable  that  several  international  institutes  shall  be 
established — at  least  four,  one  in  the  United  States,  one  in  Ar- 
gentina, one  in  Europe  or  Asia,  and  one  in  Africa  or  Australia— 
which  shall  cooperate  upon  a  fixed  general  plan,  utilize  selected 
data,  and  treat  the  subject  from  the  world-wide  point  of  view. 
The  long-range  system  will  begin  with  annual  forecasts,  already 
easy  in  Argentina,  and  advance  to  details  in  accordance  with 
the  progress  in  solar  physics.  Such  international  institutes 
must  be  equipped  with  the  highest  type  of  apparatus,  continu- 
ally improved  as  conditions  permit. 

The  progress  in  solar  physics  has  been  retarded,  as  far  as 
concerns  meteorology,  by  the  insistence  upon  immediate  fore- 


238  A  TREATISE  ON  THE  SUN'S  RADIATION 

cast  results,  before  the  fundamental  laws  have  been  classified. 
This  demand  has  already  made  it  unnecessarily  difficult  to 
obtain  the  facilities  requisite  to  work  gut  the  problems.  It 
should  be  remembered  how  small  is  the  margin  of  success  in 
short-range  forecasts,  with  all  the  extensive  coordination  that 
the  national  services  actually  possess.  In  the  United  States 
the  annual  amount  of  fair  weather  averages  about  seventy-five 
per  cent  and  the  rain  twenty-five  per  cent.  Of  this  twenty- 
five  per  cent  of  rain,  for  predictions  at  fixed  12-hour  intervals 
in  advance,  12  to  24,  24  to  36,  36  to  48  hours,  the  maximum  suc- 
cess is  about  33  per  cent,  making  83%  the  average  actual  fore- 
cast; similarly  with  the  temperature.  All  the  percentages 
above  that  amount,  that  is,  a  gain  of  only  eight  per  cent,  are  due 
to  elastic  rules  of  verification.  These  are  changed  from  time 
to  time,  but,  in  fact,  there  has  been  no  real  improvement  in  the 
past  thirty  years.  In  Argentina  the  fair  weather  percentage 
is  higher,  and  the  corresponding  average,  also,  higher. 

The  real  utility  of  weather  services  is  in  the  general  clima- 
tological  records  of  rainfall  and  temperature,  and  in  an  occa- 
sional important  forecast  of  storm  conditions,  frost  conditions, 
and,  especially,  in  the  educational  propaganda.  The  scientific 
development  has  been  entirely  sporadic,  depending  on  some  in- 
dividual initiative.  As  matters  stand,  quite  aside  from  the 
idea  of  international  cooperation  by  central  institutes,  there  is 
still  a  very  wide  margin  for  gain  in  the  theory  and  practice  of 
long-range  forecasts.  (1)  Up  to  the  present  time  meteorology 
has  had  no  central  unifying  analysis.  The  investigations  have 
been  based  upon  the  adiabatic  formulas,  which  are  wholly  in- 
applicable in  the  non-adiabatic  atmospheres  of  the  earth  and 
sun.  It  cannot  be  doubted  that  a  sound  central  theory  will 
have  the  same  good  influence  upon  atmospheric  thermodynamics, 
that  Newton's  gravitation  and  Kepler's  laws  of  motion  had 
upon  the  primitive  astronomy.  (2)  In  the  matter  of  solar  radia- 
tion the  amount  available  is  not  1.94  calories,  as  has  been  claimed, 
but  5.85  calories,  a  margin  of  200%  to  the  good  for  future 
developments.  (3)  The  method  of  distributing  meteorology, 
radiation,  magnetism,  and  solar  physics,  among  distinct  and 


RECONCILIATION    OF    DATA  239 

independent  institutions,  is  on  the  same  level  of  efficiency  as 
an  army  would  be  to  repel  invasion  with  the  infantry  in  New 
York,  the  artillery  in  New  Orleans,  the  cavalry  in  San  Francisco, 
and  the  commissariat  in  Chicago.  (4)  The  value  of  long- 
range  forecasts  is  very  great  economically,  and  the  practical 
ability  to  gain  8  or  10%  on  fair  weather  forecasts  is  not  so 
forbidding,  as  may  be  supposed.  Until  the  possibilities  of 
science  have  been  exhausted,  there  is  not  the  least  ground 
for  withholding  men  and  money  from  such  an  enterprise.  Unless 
the  government  services  approach  this  subject  in  an  efficient 
way,  there  can  be  no  doubt  that  the  demands  of  commerce 
and  agriculture  will  seek  for  other  channels  of  development, 
which  will  extend  the  range  of  the  forecasts  far  beyond  their 
present  limits. 

It  should  be  noted  that  the  small  range  of  the  solar  and 
terrestrial  synchronisms  of  the  radiation,  and  its  various 
manifestations,  culminates  in  the  case  of  the  precipitation  in 
Argentina  by  registering  about  40%  of  the  total  as  an  annual 
variation.  This  is  enough  to  distinguish  good  and  bad  agri- 
cultural years,  and  fully  justifies  all  the  cost  of  the  research. 
When  this  is  perfected  it  will  be  of  prime  value  to  the  national 
interests.  In  the  United  States  the  registration  approaches 
30%  of  the  total  in  some  localities. 


CHAPTER  VII 

Other  Solar  Phenomena 

Restatement  of  the  General  Line  of  Argument 

BEFORE  entering  upon  a  brief  discussion  of  the  other  solar 
phenomena,  following  from  the  results  of  the  thermodynamic 
v  relations  which  have  been  developed  on  the  sun,  it  is  proper  to 
restate  the  general  line  of  argument  that  has  been  explained 
regarding  the  solar  radiation.  It  has  been  shown  that  all 
gases  of  atmospheres  open  to  space  on  one  side,  but  resting 
upon  a  solid  globe  like  the  earth,  or  a  viscous  core  like  the 
sun,  possess  the  same  thermodynamic  configuration  for  the 
elements  of  the  Boyle- Gay  Lussac  law,  P  =  p  R  T.  These 
are  illustrated  in  Figures  2-13.  Near  the  bottom  of  the  iso- 
thermal layer,  where  the  temperature  gradient  suddenly  changes 
from  the  adiabatic  rate,  there  is  a  conversion  of  the  chemical 
elements  from  their  electron-ion-atomic  state  into  their  elec- 
tron-ion-molecular configuration.  During  this  process  there 
occurs  a  powerful  readjustment  of  the  electric  charges,  accom- 
panied by  numerous  rapid  changes  in  the  velocities  of  the  elec- 
trons, the  electromagnetic  oscillations  within  the  gas,  and 
release  of  plane  electromagnetic  waves  into  space.  The  thermo- 
dynamic elements  fix  the  temperature  of  the  radiation  for  all 
elements  at  about  7655°,  and  the  evidence  is  clear  that  such 
radiation  is  primarily  of  black  energy,  including  all  the  wave 
lengths  in  the  spectrum.  This  energy  passes  through  deep 
layers  of  superincumbent  gas,  and  the  electromagnetic  energy 
suffers  its  first  depletion  in  passing  over  from  electric  oscilla- 
tions into  thermodynamic  collisions  and  free  paths,  whereby 
the  equivalent  value  in  gram  calories  per  cm.2  per  minute  at 
the  distance  of  the  earth  changes  from  5.85  calories  to  3.98 
calories.  The  latter  amount  is  determined  as  that  which  is 

240 


OTHER  SOLAR  PHENOMENA  241 

received  at  the  earth,  since  the  computation  of  it  within  the 
solar  gases  involves  many  physical  terms  and  coefficients  that 
are  still  difficult  of  assignment.  The  loss  of  brightness  between 
the  center  and  the  limb  conforms  to  this  1.87  calories.  The 
determination  of  the  terrestrial  intensity,  3.98  calories,  depends 
upon  three  different  lines  of  research. 

I.  The  observed  ordinates  of   the  thermal  spectrum  by 
means  of  the  bolometer  are  best  satisfied  by  several  different 
effective  temperatures,  such  that  the  wave  lengths  1.50  ju-  2.50  /z 
originate  at  about  7655°,  while  the  wave  lengths  of  0.00  ju  to 
1.50  M  belong  to  a  complex  spectrum  energy  of  6950°  with 
certain  depletions. 

II.  Such  a  radiant  energy  in  the  space  through  which  the 
earth  passes  in  its  orbit  is  competent  to  generate  certain  tem- 
peratures in  the  earth's  atmosphere,  in  equilibrium  with  the 
kinetic  energy,  potential  energy,  inner  energy  at  the  specific 
heat  of  constant  volume,  and  the  inner  energy  at  the  specific 
heat  of  constant  pressure  as  required  by  the  force  of  gravita- 
tion.     Such   conditions   of   equilibria   are   built   up   with   the 
lapse    of    time;    they    are   independent   of    the    direction    of 
the  radiant  ray,  whether  reflected  at  the  earth's  surface  or 
direct  from  space;    and  they  are  accompanied  by  such  pres- 
sures, densities,  thermal  coefficients  as  are  determined  by  ob- 
servations between  the  sea  level  and  the  vanishing  plane  of  the 
atmosphere.     The  several  lines  of  thermodynamic  computations 
all  converge  upon  the  value  3.98  calories  for  the  effective  in- 
tensity of  the  solar  radiation  at  the  distance  of  the  earth,  thus 
agreeing   with   the   bolometer   ordinates   between   0.00  M   and 
IJJO'ji. 

III.  The  pyrheliometer  measures  the  kinetic  energy  of  radia- 
tion in  temperature-degrees  at  the  distance  of  the  sun  from  the 
zenith,  and  by  the  Bouguer  formula  its  value  can  be  found 
in  the  zenith,  together  with  the  coefficient  of  absorption  near 
the  station.     In  the  computations  the  corresponding  air-poten» 
tial  energy,  the  corrections  for  the  specific  heat,  the  depletion 
by  scattering  and  by  absorption,  can  be  supplied,  and  these 
again  lead  to  3.98  calories  at  stations  in   the  northern   and 


242  A  TREATISE   ON  THE  £UN's   RADIATION 

the  southern  Americas.  The  Langley-Abbot  method  of  reduc- 
ing the  pyrheliometer  observations  involves  a  number  of  assump- 
tions and  omissions  which  invalidate  it. 

1.  Extrapolation  to  the  non-mathematical  sec  2  =  0. 

2.  Assumption  that  sec  z  —  1  necessarily  is  identical  with 
the  depth  of  the  earth's  atmosphere. 

3.  Assumption  that  the  line  and  band  depletions  used  in 
making  up  the  bolometer  factor  should  be  referred  to  the  spec- 
trum of  6000°,  instead  of  6950°,  which  is  purely  arbitrary. 

4.  Omission  of  the  potential  energy  term. 

5.  Omission  of  the  specific  heat  term. 

6.  Omission  of  the  ionization  energy  term.     Removing  these 
defects   the   pyrheliometer   data   become   accordant   with   the 
bolometer  data,  and  the  thermodynamics  of  the  atmospheres 
of  the  earth  and  the  sun. 

The  terrestrial  meteorological  data  are  much  better  satisfied 
practicably  by  reference  to  a  radiation  energy  of  about  4.00 
calories  than  they  are  by  that  of  2.00  calories.  In  fact,  it  is 
impracticable  to  build  up  the  prevailing  temperatures,  pressures, 
and  densities,  together  with  their  climatological  changes,  by 
employing  Abbot's  solar  intensity  of  1.930  calories. 

We  shall  now  proceed  to  make  further  use  of  the  solar  thermo- 
dynamic  data  in  reference  to  the  origin  of  the  sun-spots,  faculae, 
prominences,  sharp  disk  of  the  sun,  apparent  diminution  of 
brightness  between  the  center  and  the  edge  of  the  sun,  the 
corona,  the  solar  magnetic  field,  the  solar  electrostatic  field, 
the  general  circulation  in  the  sun's  mass.  Each  of  these  sub- 
jects requires  a  prolonged  research  for  developing  its  details, 
so  that  only  a  short  descriptive  statement  will  be  practicable 
in  this  chapter. 

The  Effect  of  the  Solar  Isothermal  Shell  upon  the  Visible  Surface 

Phenomena 

The  most  prominent  fact  regarding  the  physical  constitu- 
tion of  the  sun  is  that  there  exists  an  isothermal  shell  of  the 
mean  temperature  of  about  7680°  located  in  the  midst  of  the 
solar  gases  at  a  fixed  distance  from  the  center  of  the  mass. 


OTHER  SOLAR  PHENOMENA  243 

This  shell  is  maintained  by  the  thermodynamic  processes  in 
this  position  by  the  interrelated  actions  of  gravitation  and 
radiation,  as  has  been  explained.  The  outer  surface  is  called 
the  photosphere,  and  it  appears  as  a  sharp  circle  on  the  disk, 
because  the  heavy  gases  vanish  near  those  levels,  and  on  ac- 
count of  their  large  number  they  produce  the  observed  optical 
brilliancy  of  the  sun.  The  lighter  gases  extend  upward  through 
the  levels  of  the  flash  spectrum,  the  chromosphere,  and  the 
inner  corona,  the  latter  being  most  conspicuous  in  hydrogen. 
Beneath  the  isothermal  layer  the  gases  recede  at  adiabatic 
rates  to  great  depths,  which  finally  coalesce  in  the  quasi-solid 
nucleus.  Compare  Section  I,  Fig.  28.  In  the  midst  of  the 
isothermal  layer  the  gases  go  through  their  thermodynamic 
development  in  strata  having  a  special  depth  for  each  gas,  and 
yet  they  are  entirely  independent  of  one  another.  The  isother- 
mal layer  is  a  complex  of  numerous  gases,  each  one  agreeing 
with  all  the  others  as  to  temperature,  while  differing  as  to  the 
vertical  scale  of  their  strata  of  equal  temperatures.  We  may, 
therefore,  having  recognized  the  very  complicated  structure  of 
the  isothermal  shell,  proceed  with  further  explanations,  sub- 
stituting the  simple  idea  of  a  single  shell  of  uniform  tempera- 
ture, at  the  mean  distance  R  =  694800800  meters  from  the 
center  of  the  sun.  The  sun-spots,  faculae,  and  spectra,  all 
testify  to  the  complex  internal  structure  of  the  solar  gases  in 
the  neighborhood  of  the  photosphere,  and  some  progress  has 
been  made  heretofore  in  classifying  these  phenomena,  though 
this  was  much  impeded  by  the  lack  of  a  definite  knowledge 
of  the  pressures,  densities,  and  gas  efficiencies  which  are  asso- 
ciated with  them. 


The  Granulations,  Facula,  Flocculi,  and  Prominences 

The  literature  regarding  the  appearance  and  origin  of  these 
solar  phenomena  is  very  extensive,  and  it  consists  of  numerous 
explanations  of  the  probable  causes  which  produced  them,  in 
accordance  with  some  observed  physical  conditions  on  the  sun. 
We  shall  have  much  advantage  in  this  respect  from  an  accurate 


244 


A  TREATISE   ON  THE   SUN^S   RADIATION 


knowledge  of  the  temperatures,  pressures,  densities,  and  gas 
efficiencies  that  prevail,  and  it  is  not  too  much  to  say  that 
approximate  computations  can  easily  be  made,  such  as  would 
reproduce  these  appearances  in  the  neighborhood  of  the  photo- 


FIG.  28.  I.  Formation  of  the  Solar  Photosphere.  II.  Formation  of  the 
Granulation,  Prominences,  and  Sun-spots.  III.  Elements  of  the  General 
Circulation  within  the  Sun.  IV.  Refraction  in  the  Solar  Atmosphere. 

sphere.  I  shall  avail  myself  freely  of  the  results  of  the  Mt. 
Wilson  Observatory,  and  of  the  numerous  papers  from  other 
authors  in  the  Astrophysical  Journal,  in  stating  the  facts  per- 
taining to  the  several  topics. 

The    granulations,    faculae,   flocculi,    and    prominences    are 


OTHER  SOLAR  PHENOMENA  245 

correlative  phenomena,  all  depending  upon  thermodynamic 
processes  in  several  levels.  The  granulation  is  a  convective 
result  on  the  top  of  the  photosphere;  the  faculae  lie  in  the 
midst  of  or  just  above  the  photosphere,  while  the  flocculi  are 
in  the  higher  levels  of  the  convective  columns;  and  the  promi- 
nences may  extend  by  eruption  far  beyond  the  normal  levels 
of  the  hydrogen  atmosphere.  The  forces  of  segregation  de- 
pend upon  the  intrusion  of  the  temperatures,  and  all  the  ele- 
ments associated  with  them,  into  levels  which  are  inconsistent 
with  the  equilibria  of  gravitation.  If  there  is  an  equilibrium, 
then  the  equation  of  condition  holds. 

(176)  G  fe  ~  So)  =  -  Pl~P°  -  *  (<?i2-  <?o2)  -  (PFi  -  Wo)  - 

Pio 


The  gravitation  acceleration  between  two  levels  balances  the 
hydrostatic  pressure,  the  kinetic  energy  of  circulation,  the  work 
of  expansion  against  external  forces,  and  the  inner  energy. 
If  these  are  not  in  equilibrium  in  the  stratum  the  temperature 
is  no  longer  normal,  the  residual  A  G  (z\  —  z0)  has  value,  and 
the  process  of  readjustment  sets  in  at  once.  The  first  step  in 
adjustment  is  a  change  in  the  velocity  of  circulation  between 
the  top  velocity  q\  and  the  bottom  velocity  %  thus  resulting 
in  an  addition  to  the  gravity,  plus  or  minus,  on  a  rotating  globe; 
and  the  second  step  is  in  a  change  of  inner  energy  through 
radiation.  The  formulas  of  Chapter  I  should  be  kept  contin- 
ually in  mind,  especially  in  respect  of  the  fact  that  no  term 
can  change  without  drawing  with  it  the  entire  long  train  of 
physical  terms  that  are  united  with  it.  Hence,  to  refer  any 
physical  solar  appearance  and  effect  to  a  single  property,  as 
pressure  or  density,  is  very  incomplete  and  even  erroneous. 
This  large  complex  of  causes  and  effects,  on  the  other  hand, 
makes  all  atmospheric  physics  difficult,  and  renders  mere  de- 
scription of  little  value.  However,  it  is  not  possible  to  trace 
out  into  its  proper  details  all  these  interrelated  terms,  but  it  is 
proper  to  insist  that  they  all  merely  illustrate  the  several  terms 
in  the  equation  in  some  of  their  phases.  Thus,  the  granulation, 


246 


A  TREATISE   ON  THE   SUN'S   RADIATION 


faculae,  and  flocculi,  as  convective  phenomena,  exhibit  more 
the  changes  in  the  inner  energy,  while  the  prominences  are 
more  closely  related  to  kinetic  circulation,  the  direction  of  the 
former  class  being  vertical  and  the  latter  horizontal.  If  gases 
are  thrown  upward  from  below  by  internal  convection,  through 
the  levels  of  the  isothermal  layer  and  the  photosphere,  it  can 
be  seen  that  they  may  pass  through  very  great  differences  in 
temperature  in  a  short  vertical  distance.  For  example,  by 
Table  6,  the  changes  can  be: 
H2  from  7670°  to  955°,  from  the  photosphere  to  20000  kilometers. 


He 

C 

Ca 

Zn 

Cd 

Hg 


7705°  to  740°, 
7695°  to  0°, 
7705°  to  300°, 
7715°  to  640°, 


7684°  to 
7670°  to 


0,° 
0,° 


10000 

5000. 

1000 

600 

400 

300 


Similarly,  compare  Table  9  for  pressure,  Table  12  for  den- 
sity, and  Table  13  for  the  gas  efficiency.  A  short  vertical  move- 
ment of  a  mass  of  gas  is  therefore  equivalent  to  an  immense 
amount  of  kinetic  energy  of  circulation.  The  same  gas  at  its 

TABLE  86 

VERTICAL  CHANGES 

Calcium  has  the  following  data 


Height 

T 

A 

P 

p 

R 

1000  Km  
700    " 

300° 
2200 

7.654X10-10 
0  003579 

7.755X10-6 
362  57 

2.948X10-8 
0  000296 

8.7702 
556  73 

400    "     . 

4700 

0  30009 

30407 

0  0042217 

1532  5 

100    "     
0    "     

7500 
7705 

2.8405 
5.3060 

287830 
537620 

0.0162620 
0.023657 

2360.1 
2949.5 

Hydrogen  has  the  following  data 


Height 

r 

A 

p 

p 

R 

20000  Km.... 

955° 

3.728X10-7 

0.03778 

8.051X10-8 

4912.8 

10000    "    

3495 

0.0806 

8165.6 

0.00005006 

46669 

5000    "    .... 

5890 

1.0687 

108265 

0.0003146 

58436 

1000    "    .... 

7680 

4.3512 

440880 

0.0008538 

67237 

0    "    .... 

7670 

5.9084 

598657 

0.0010613 

73547 

Note  the  effects  of  vertical  motion  on  T.  A .  P.  p.  R. 


OTHER  SOLAR  PHENOMENA  247 

different  levels  has  such  a  wide  range  of  temperature  for  radia- 
tion and  absorption  that  it  may  be  able  to  reverse  its  spectrum 
several  times. 

There  are  certain  bands  of  calcium,  H  .  K,  which  are  seen 
to  have  a  double  reversal,  corresponding  to  the  successive  rela- 
tive emissions  and  absorptions  at  different  heights.     They  are 
classified  by  Mr.  Hale: 
H3 .  K3,  narrow  line  of  high  level  absorption  at  700  to  1000 

kilometers. 

H2 .  K2,  middle  lines  of  emission  at  400  to  700  kilometers. 
HI.  KI,  wide  base  of  band  of  absorption  at  100  to  400  kilometers. 
H  .  K,  low  level  continuous  spectrum  at  0  to  100  kilometers. 

It  would  seem  that  a  change  in  temperature  of  about  2000 
degrees  is  sufficient  to  make  the  successive  steps  in  alternate 
emission  and  absorption  from  one  level  to  another.  In  the 
case  of  calcium  the  vertical  interval  is  not  far  from  300  kilom- 
eters; in  the  case  of  hydrogen  Ha,  H  0,  H  7,  they  are  4000 
to  5000  meters,  increasing  upward.  Calcium  has  the  molecular 
weight  40,  and  it  is  located  on  the  hyperbolic  curves  of  Fig.  18 
where  the  rapid  rise  to  the  vertical  branch  begins.  Hydrogen 
extends  upward  to  the  maximum  height.  The  elements  be- 
tween these  two  enter  freely  into  all  the  visible  surface  phe- 
nomena, while  those  elements  below  m  =  40  enter  more  diffi- 
cultly into  these  appearances.  This  capacity  of  vertical  exten- 
sion naturally  divides  the  convectional  columns  into  separate 
classes,  though  they  merge  into  one  another.  Thus,  the  flocculi 
are  the  high-level  columns  of  the  light  gases  seen  in  section 
by  means  of  the  spectroheliograph  only;  the  faculce  are  low- 
level  sections  just  above  the  photosphere,  as  in  the  lower  parts 
of  the  calcium  columns;  the  granulations  are  the  heads  of  the 
columns  on  the  level  of  the  photosphere,  and  consist  chiefly 
of  the  heavy  gases.  All  of  the  strata  of  the  isothermal  layer, 
and  of  the  non-adiabatic  layer,  are  more  or  less  segregated  into 
columns  of  different  temperature  and  density  in  vertical  direc- 
tions, and  these  extend  to  different  altitudes  according  to  their 
thermodynamic  conditions.  One  of  the  important  features  of 
atmospheres  is  that  masses  of  the  same  gas  having  different  tern- 


248  A  TREATISE  ON  THE   SUN'S  RADIATION 

peratures  and  densities  are  reluctant  to  mix  thoroughly  with  one 
another.  Such  masses  will  maintain  themselves  side  by  side, 
standing  vertically  on  a  common  base,  or  lying  alongside  each 
other  on  the  same  horizontal  plane,  for  long  intervals  without 
losing  their  individuality.  Such  juxta-masses  interchange  their 
thermodynamic  values  slowly  at  the  surfaces,  and  the  shell  of 
mutual  equilibrium  only  penetrates  gradually  to  the  middle  of 
large  masses.  For  this  reason  circulation  sets  in  as  minor  vor- 
tices and  mixing  currents,  which  form  a  sheath  of  kinetic  energy 
on  the  surface  of  the  mass,  whereby  the  atomic-molecular  equi- 
libria may  be  transmitted  in  all  their  details.  This  consumes 
time,  and  in  the  case  of  large  masses  whose  temperature  par- 
ameters are  not  very  far  apart,  this  may  be  prolonged;  where 
the  temperatures  differ  widely  the  kinetic  movements  integrate 
into  general  movements  of  the  entire  /masses.  These  move- 
ments may  be  vertical,  as  where  convection  currents  rise  and 
fall  in  the  immediate  neighborhood  of  one  another,  or  they 
may  be  horizontal,  as  where  currents  of  different  temperatures 
flow  along  horizontal  planes.  These  have  the  property,  by 
means  of  the  term  |  (#i2  —  <?02),  to  add  to  or  subtract  from 
the  local  pressure  and  density  terms,  so  that,  under  the  force 
of  gravitation,  these  shall  be  preserved  from  abnormal  curva- 
tures on  the  respective  levels.  Some  account  of  "this  process 
is  given  in  the  "Meteorological  Treatise,"  but  the  subject  is  one 
of  difficulty  because  of  the  complexity  of  the  details  of  the 
thermodynamics . 

These  explanations  are  necessary  to  be  remembered  as  the 
causes  of  the  solar  segregations  of  gases.  The  common  descrip- 
tive solar  physics  is  a  long  chapter  of  these  phenomena,  but 
they  all  illustrate*  these  thermodynamic  results.  On  the  other 
hand,  it  will  probably  be  possible  to  proceed  from  the  observed 
results  through  the  formulas  to  the  primitive  energies  of  circu- 
lation and  radiation  that  are  involved.  This  juxtapostion  of 
masses  of  discontinuous  temperatures,  in  vertical  and  in  hori- 
zontal planes,  gives  rise  to  anomalies  in  refraction,  and  allied 
phenomena.  Some  further  account  of  solar  atmospheric  refrac- 
tion will  be  added  on  a  later  page. 


OTHER  SOLAR  PHENOMENA  249 

Granulations.  The  surface  of  the  photosphere,  as  seen  di- 
rectly in  a  telescope,  seems  to  consist  of  bright  and  dark  patches 
of  irregular  forms.  They  are  called  "  fleecy  clouds,"  "  fleecy 
patches,"  "  rice  grains,"  "  granules,"  and  they  are  in  their  maxi- 
mum large,  and  in  their  minimum  they  have  been  observed 
down  to  2".6,  1".4,  or  even  to  0".3  in  diameter.  On  the  sur- 
face of  the  sun,  whose  diameter  =  1389602  kilometers,  or 
1919".26,  we  have  1"  =  724.03  kilometers  =  449.89  miles. 
Hence,  we  have,  2".6  =  1882  kilometers  =  1170  miles;  1".4  = 
1014  kilometers  =  630  miles;  and  0".3  =  217  kilometers  =  135 
miles.  The  finest  estimated  filaments  are  0".03  =  21.7  kilom- 
eters =  13.5  miles.  We  must  infer  that  the  filamentary  or 
threadlike  structure  has  no  real  limit  in  its  tenuity,  either  in  ver- 
tical or  in  horizontal  directions.  The  segregations  may  appear 
large,  but  they  are  interpenetrated  by  innumerable  vortex  lines 
of  complex  form,  and  the  individuality  in  some  degree  depends 
upon  more  or  less  perfect  vortex  motions.  These  interpene- 
trations  may  culminate  in  a  snarl  of  vortex  actions,  with  con- 
fused temperatures  and  densities,  favorable  to  the  production 
of  diffuse  reflection  and  scattered  radiation  in  their  midst. 
The  tops  of  the  vertical  filaments  seen  in  section  are  evidences 
of  such  action,  and  the  bright  patches  of  granulation  on  the 
photosphere  are  thus  constructed.  There  are  two  causes  for 
their  perpetual  existence  on  that  level:  (l)  The  heavy  metals 
or  "gases  terminate  near  that  level,  so  that  vertical  convection 
alone  will  place  their  heads  just  above  the  isothermal  levels; 
(2)  Since  the  gradients  of  temperature  are  nearly  horizontal  for 
the  heavy  gases,  and  these  must  penetrate  through  the  tem- 
perature gradients  of  the  lighter  gases,  there  is  a  stratum  of 
very  pronounced  optical  confusion  just  along  these  levels,  due 
to  the  mixture  of  the  horizontal  and  vertical  temperature  gra- 
dients of  the  heavy  and  the  light  gases  respectively. 

This  interpenetration  is  readily  perceived  by  Figs.  10,  11, 
12,  and  Fig.  28,  Section  I,  when  it  is  remembered  that  every 
unit  volume  of  the  isothermal  and  lowest  non-adiabatic  strata, 
all  over  the  spherical  shell  of  the  sun,  is  full  of  such  a  mixing 
series  of  atmospheres.  Within  the  isothermal  layer  the  fila- 


250  A  TREATISE   ON  THE  SUN'S  RADIATION 

ment-structure  is  generally  vertical  without  serious  disturbance, 
but  above  this  stratum  there  is  a  strong  tendency  to  horizontal 
filaments.  The  combination  may  produce  curved  filaments  of 
every  possible  shape,  such  as  are  seen  in  sun-spots,  prominences, 
and  faculae.  Above  this  region  of  special  mixture,  the  filaments 
tend  to  become  vertical  once  more  throughout  the  chromo- 
sphere and  inner  corona,  which  consist  of  light  gases.  The  irreg- 
ularities of  mixture  there  take  fantastic  forms,  more  or  less 
stationary,  and  in  the  prominences  the  cooling  sheath  of  mixture 
on  the  edges  of  the  mass  of  gas  has  been  repeatedly 
remarked. 

FaculcB.  These  are  the  irregular  large  regions  of  vertical 
convection,  closely  associated  with  the  region  of  the  sun-spots 
and  their  production,  and  they  are  the  evidences  of  a  general 
convective  movement  from  the  interior  of  the  sun  on  a  large 
scale.  They  are  seen  most  conveniently  in  the  calcium  lines, 
since  it  is  very  abundant,  and  has  a  broad  base  with  rapidly 
diminishing  density  to  1000  kilometers.  The  other  light  gases 
have  similar  distributions  so  far  as  they  can  be  observed. 

Flocculi.  These  are  sectional  aspects  of  the  vertical,  con- 
vectional  columns  in  the  light  gases  at  greater  altitudes,  and 
in  calcium,  hydrogen,  carbon,  they  are  most  readily  observed. 
The  great  difference  in  the  altitudes  possible  for  calcium  and 
hydrogen  makes  evident  their  respective  positions.  The  areas 
of  the  sections  of  the  columns  increase  with  the  height,  as  the 
convectional  vortex  spreads  out  horizontally.  Every  rising 
column  tends  to  throw  off  lateral  vortex  whirls  along  its  sides, 
and  the  juxtaposition  of  such  vortices  produces  mixture,  while 
they  develop,  also,  downward  return  currents.  The  crests  of 
the  heavy  gases  become  the  heads  of  the  granulation,  which 
are  bright,  but  the  colder  downward  currents  between  them 
are  relatively  dark.  Likewise,  the  faculae  and  flocculi  produce 
similar  curling  vortices  with  crests,  mixing  sides,  and  downward 
interspacial  currents.  Compare  Fig.  28,  Section  II,  A.  B. 

Prominences.  These  are  the  hydrogen  flames,  seen  in  a 
spectroscopic  slit  on  the  limb  of  the  sun,  and  they  belong  to 
the  so-called  eruptive  type  of  convection,  whereby  the  thermo- 


OTHER  SOLAR  PHENOMENA  251 

dynamic  forces  of  the  lower  levels  develop  great  spacial  ex- 
pansions when  transported  upward,  on  account  of  the  small 
density  of  the  hydrogen  in  the  high  strata.  The  vertical  col- 
umns are  easily  observed,  and  the  curls,  more  or  less  pronounced, 
are  recognized.  There  are  large  apparent  motions,  such  as 
100  miles  per  second,  and  this  great  apparent  velocity  may  be 
due  to  the  translation  of  masses,  or  to  the  propagation  of  optical 
effects  by  successive  local  changes  in  density  and  scattering 
through  mixtures.  Large  masses  may  become  visible  by  sur- 
face changes  of  the  thermal  conditions,  and  in  many  cases  this 
accounts  for  great  apparent  velocities  of  propagation.  To  dis- 
tinguish between  these  two  processes  is  not  easy,  at  the  distance 
of  the  sun,  under  the  complex  conditions  which  prevail,  involv- 
ing the  great  force  of  gravitation  in  the  equilibrium  with  the 
other  thermal  terms  of  the  equation  of  condition. 

The  Origin  of  the  Sun-Spots 

The  literature  of  solar  physics  abounds  in  suggestions  and 
hypotheses  regarding  the  significance  and  the  origin  of  the  sun- 
spots  and  their  attendant  phenomena.  These  are  based  upon 
the  different  features  that  are  observed,  each  theory  empha- 
sizing some  physical  characteristic,  and  doubtless  there  must 
be  elements  of  truth  in  nearly  every  one  of  the  leading  schemes. 
Yet,  it  is  commonly  believed  that  the  subject  is  still  open  to 
research,  and  that  something  has  been  lacking  capable  of  har- 
monizing the  facts  derived  from  direct  telescopic  and  spectro- 
scopic  observations.  While  the  analysis  of  the  solar  data  given 
in  the  previous  pages  may  be  quite  sufficient  to  open  up  new 
paths  of  research  and  to  reconcile  the  several  outstanding  dif- 
ficulties, it  is  probable  that  further  experience  will  enable  us 
to  make  still  further  advances,  especially  in  the  interpretation 
of  the  thermal  and  the  optical  spectra.  It  is  impracticable  to 
review  carefully  the  literature  regarding  sun-spots,  but  enough 
may  be  summarized  to  bring  out  the  most  important  facts 
that  must  be  taken  into  the  account. 

Herschel  conceived  that  there  are  three  distinct  layers  in  the 
solar  constitution:  a  solid  nucleus  seen  as  umbra,  a  quasi-liquid 


252 


A  TREATISE   ON  THE   SUN'S  RADIATION 


layer  seen  as  penumbra,  and  a  luminous  envelope  of  condensa- 
tion forming  the  cloudy  photosphere. 

Secchi,  Faye,  and  many  others,  make  the  spots  the  centers 

Non-adiabatic  Region 
Chromosphere^v.      \      and     /         Reversing  Layer 

Cold          Faculae~~K^>>\\    |   / /^fr^LSTr**       Cold 
^s 


FIG.  29.     General  Scheme  of  the  Structure  of  a  Sun-spot  Vortex,  and  Its 
Relations  to  the  Reversing  Layer.     (Bigelow.) 

of  explosive  outbursts  of  imprisoned  gases  retained  within  a 
quasi-liquid  restraining  envelope. 

Secchi,  Lockyer,  Oppolzer,  and  others,  consider  the  sun-spots 
the  center  of  downpouring  gases,  or  meteoric  material,  bringing 
cooler  temperatures  from  the  higher  strata.  -  Young  sees  in  the 
spot  an  area  of  sinking  of  the  photosphere  and  elevation  of  a 
ring  surrounding  it.  Fox,  and  others,  observe  that  spot-birth 
is  always  preceded  by  an  eruption  of  faculae  in  a  ring  outside 
of  the  penumbra,  most  pronounced  on  the  following  side. 

Schmidt  applies  circular  refraction  to  make  the  spot  a  mirage 
effect,  while  the  phenomenon  really  occurs  at  considerable  depths 
below  the  photosphere. 

Julius  applies  the  optical  effects  of  anomalous  dispersion 
and  complex  refraction  to  the  entire  series  of  spot  and  attendant 
phenomena. 

Schaeberle  finds  mechanical  forces  enough  to  account  for  the 
conditions. 


OTHER  SOLAR  PHENOMENA  253 

Brester  applies  chemical  luminescence  and  moving  flashes, 
among  other  phenomena  of  that  kind. 

Sidgreaves  utilizes  the  chill  of  expansion  to  account  for  the 
low  temperature  in  the  central  areas  of  spots. 

Hale,  Adams,  and  others,  find  that  the  spectra  indicate  an 
almost  universal  weakening  of  the  enhanced  lines  in  sun-spots, 
and  consider  this  as  evidence  of  the  low  temperature  in  the 
reversing  layer  over  the  spots  themselves. 

Abbot  determines  lower  temperatures  in  the  spots  than  in 
the  surrounding  photosphere  by  bolometer  observations. 

The  observers  of  the  Mt.  Wilson  Observatory  have  measured 
in  the  displacement  of  the  spectrum  lines  that  there  are  out- 
ward velocities  in  the  lower  part  of  the  sun-spot  region  and  in- 
ward velocities  in  the  upper  part.  There  are  evidences  of  a 
slow  spiral  inward  movement  from  the  photosphere  at  consid- 
erable distances  into  the  penumbra,  and  thence  downward  to 
the  umbra. 

Langley  pointed  out  that  the  solar  surface  is  striated  in 
vertical  columns,  like  sheaves  of  wheat,  that  the  quiescent  tops 
of  the  segregations  appear  as  granules,  and  that  when  bent 
over  they  form  the  filaments  in  the  penumbra. 

Faye  tried  to  account  for  a  solar  vortex  as  a  whirlpool  in  a 
stream  having  relatively  different  eastward  velocities. 

Bigelow  wrote  as  follows  in  respect  of  the  connection  between 
solar  vortices  and  the  dumb-bell-shaped  vortex,  Monthly 
Weather  Review,  October,  1908: 

"The  sun-spots  occur  on  the  outer  surface  of  the  photo- 
sphere and  extend  inward  toward  the  center  of  the  sun.  They 
consist  visibly  of  a  nucleus  which  is  practically  structureless, 
and  a  penumbra  which  is  striated  radially  with  much  regularity. 
The  observed  movements  of  the  material  composing  the  penum- 
bra are  from  the  outer  edge  of  the  disturbed  area  in  the  photo- 
sphere toward  the  umbra,  and  the  radial  striae  usually  termi- 
nate in  ends  which  are  bent  downward  toward  the  interior  of 
the  sun.  The  motion  of  a  particle  starting  on  the  outer  edge 
of  the  penumbra  is  primarily  inward,  and  then  rather  suddenly 
downward.  This  corresponds  so  closely  to  the  motion  in  the 


254  A  TREATISE   ON  THE   SUN'S   RADIATION 

upper  levels  of  a  dumb-bell-shaped  vortex  where  the  circulation 
is  downward  that  it  seems  proper  to  suggest  this  explanation 
of  the  origin  and  structure  of  the  sun-spots.  The  sun-spots 
would  correspond  to  the  layers  between  the  sections  a  z  =  180° 
and  az  =  170°  (M.  W.  R.,  Oct.  1907,  p.  475,  fig.  3),  if  the  cir- 
culation is  downward.  In  this  limited  region  there  is  practically 
little  rotary  velocity  v,  the  vertical  velo'city  w  becomes  important 
only  when  approaching  the  abrupt  curvature,  which  is  here 
assumed  to  be  on  the  outer  edge  of  the  umbra,  but  in  the  pe- 
numbra the  radial  velocity  is  conspicuous.  The  sun-spot  may 
be  caused  by  layers  of  matter  inside  the  sun's  photosphere  oper- 
ating to  draw  material  downward,  warm  layers  being  super- 
posed upon  cold  layers  at  the  section  which  corresponds  with 
the  lower  plane  of  the  sun-spot  vortex." 

Comparable  ideas  are  quoted  by  C.  E.  St.  John,  from  Moore's 
"  Meteorology,"  in  his  paper  on  the  radial  motion  in  sun-spots, 
Astrophysical  Journal,  Vol.  XXXVII,  1913.  This  paper  pre- 
sents certain  values  of  the  outward  and  the  inward  radial  veloci- 
ties u  in  different  levels,  as  derived  from  the  displacement  spec- 
trum lines.  Generally,  the  sun-spot  is  now  considered  to  be  a 
complex  vortex,  and  we  shall  proceed  to  utilize  our  solar  data 
in  order  to  examine  more  closely  its  probable  origin,  and  struc- 
ture. It  will  be  seen  that  my  views  of  1908  have  been  modified 
in  details  in  consequence  of  this  new  data. 

It  should  be  remembered,  as  explained  in  my  "Meteorological 
Treatise,"  that  Nature  abhors  a  discontinuity  in  the  curves  of 
pressure  and  density,  and  that  temporary  abnormalities  in  these 
curves,  caused  by  imperfect  thermodynamic  adjustments,  are 
compensated  by  horizontal  currents  in  general.  Since  a  hori- 
zontal current  transfers  its  energy  into  a  vertical  force,  by  the 
general  equations  of  motion  it  follows  that  these  currents  occur 
at  the  boundary  of  cool  and  warm  strata,  whenever  the  tem- 
perature gradient  becomes  abrupt  and  abnormal.  In  the  solar 
atmosphere  a  small  convectional  lift  of  the  adiabatic  strata,  at 
the  lower  boundary  of  the  isothermal  stratum,  produces  at  once 
a  horizontal  outflow  from  the  center  or  point  of  greatest  dis- 
continuity in  T.  The  lower  planes  of  the  several  isothermal 


OTHER  SOLAR  PHENOMENA  255 

strata  are  therefore  the  planes  of  vortical  outflow,  so  long  as 
the  internal  vertical  convection  from  the  interior  of  the  sun 
persists.  Since  all  gases  must  pass  through  the  transformation 
from  adiabatic  to  isothermal  temperatures,  that  is  from  large 
negative  gradients  to  small  positive  gradients,  it  is  evident  that 
the  mass  of  gas  containing  —  (Qi  —  QQ)  can  not  dispossess  itself 
of  this  heat  content  with  sufficient  rapidity  to  conform  to  the 
change  in  temperature  gradient  without  other  special  processes. 
One  of  these  has  been  described,  namely,  the  generation  of  the 
solar  radiation,  and  the  other  is  the  production  of  the  sun-spot 
vortices.  Hence,  concentrations  of  radiation  and  of  sun-spot 
phenomena  happen  wherever  the  internal  thermodynamics 
within  the  enormous  mass  of  the  sun,  operating  by  its  own 
laws,  may  determine  its  great  currents  of  general  circulation. 
These  concentrations  may  occur  in  latitude,  as  on  the  maxima 
of  the  sun-spot  belts,  or  in  longitude,  as  indicated  by  the  suc- 
cessive maxima  and  minima  along  the  equator,  as  heretofore 
found  to  be  the  case  by  several  lines  of  evidence.  The  general 
result  is  to  produce  a  series  of  spherical  harmonics,  and  there 
is  little  doubt  that  their  characteristics  can  be  worked  out  from 
our  materials. 

Since  the  same  sun-spot  endures  for  months  on  occasions, 
it  is  necessary  to  determine  not  only  what  are  the  forces  which 
originate  them,  but  what  physical  process  sustains  them  during 
such  long  intervals  of  time.  This,  in  connection  with  the  minor 
features  enumerated  above,  is  the  object  of  our  research.  The 
discovery  of  the  permanent  existence  of  a  spherical  isothermal 
shell  around  the  sun,  of  persistent  thermodynamic  origin,  greatly 
facilitates  our  effort.  It  is  seen  that  all  of  the  gases  which 
by  vertical  convection,  either  from  below  upward  or  from 
above  downward,  attempt  to  pass  through  this  layer,  are 
necessarily  reduced  to  this  isothermal  temperature,  between 
7650°  and  7700°,  whatever  may  be  their  temperature  on 
approaching  this  shell.  For  brevity,  we  shall  continue  to  speak 
of  the  parameter  T  as  implying  the  long  train  of  thermodynamic 
processes  that  have  already  been  described,  including  associa- 
tion and  dissociation,  radiation  and  absorption,  dynamic  move- 


256 


A   TREATISE   ON  THE   SUN'S   RADIATION 


ments  and  thermodynamic  transformations.  It  is  especially  to 
be  remarked  that  in  the  adiabatic  strata,  below  the  isothermal 
layer,  a  short  vertical  movement  of  a  mass  of  gas  by  convec- 
tion changes  its  level  of  temperature  through  many  degrees, 
and,  similarly,  after  passing  the  isothermal  layer,  there  is  a  fur- 
ther great  change  of  temperature  with  the  height.  This  is  en- 
tirely true  of  the  heavy  gases,  and  progressively  less  true  of  the 
light  gases. 

TABLE  87 
VERTICAL  DISTANCE  REQUIRED  FOR  A  CHANGE  OF  1000° 


Element 

Above  the 
Isothermal  Layer 

Below  the 
Isothermal 
Layer 

Hydrogen,  HI  1  
Hydrogen,  H2  2  
Helium,  He  4  
Carbon,  C  12  

6630  (' 

4200 
1700 
620 
145 
85 
56 
30 

? 

Hi 

4360  (, 
1940 
980 
340 
82 
68 
28 
20 

ft 

3 

CO 

2100  K 
1450 
500 
180 
53 
33 
21 
11 

m.  (4) 

1  to 

sll 

Calcium,  Ca  40  

Zinc,  Zn  65  
Cadmium,  Cd  112. 

Mercury,  Hg  198  

Table  87  collects  these  vertical  temperature  gradients  into 
three  classes,  as  from  0°  to  4000°  marked  (2),  5000°  to  7000° 
marked  (3)  above  the  isothermal  layer,  and  8000°  to  12000° 
marked  (4)  below  it  in  the  adiabatic  region.  The  change  of 
gradient  is  progressive  so  that  the  table  is  merely  illustrative. 

.      6330    3600    2150     ,  ,  , 

The  ratios  ,  ,  ,  show  how  to  proceed  from  one 

m        m        m 

element  to  another  through  the  atomic  weights.  Evidently 
short  vertical  changes  for  the  heavy  gases  are  matched  by 
long  vertical  changes  for  the  light  gases. 

In  order  to  understand  the  significance  of  a  vertical  convec- 
tive  movement,  we  require  the  general  equation  of  condition, 
including  the  kinetic  energy  of  circulation, 


G  (Zi  -  So)   =    - 


Pio 


-  -A)  -  (Qi  -  Co). 


On  a  level  of  equilibrium  without  circulation,  the  term  —  i 
(q\  —  qzo)  disappears,  but  if  by  convection  the  mass  containing 


OTHER  SOLAR  PHENOMENA 


257 


—  (Qi  —  Qo)  free  heat  is  moved  to  a  higher  level,  the  equilibrium 
is  destroyed  temporarily,  and  in  the  process  of  adjustment  the 
term  —  J  (q\  —  qzo)  is  set  in  operation,  and  it  will  be  maintained 
as  long  as  continuous  vertical  convection  renews  the  —  (Qi  —  QQ) 
Sit  the  abnormal  level.  Computing  the  available  velocity,  we 
have  for  different  heights — that  is,  for  different  discontinuities — 
of  the  thermodynamic  levels: 

TABLE  88 

HORIZONTAL  VELOCITY  FOR  VERTICAL  CONVECTION  TO  DIFFERENT 

HEIGHTS 


Change  in  Height 

H2 

He 

t  c 

Ca 

Zn 

Cd 

Hg 

Means 

50  Kilometers  
5    "    

1484 
469 

1581 
500 

1703 
539 

1673 
529 

1483 
469 

1658 
524 

1597 
505 

1597 
505 

1    "    

210 

223 

241 

237 

210 

234 

226 

226 

If  the  mass  containing  the  free  heat,  as  computed  at  the 
top  of  the  adiabatic  levels,  or  in  the  lower  part  of  the  isothermal 
stratum,  is  raised  50  kilometers*,  it  can  be  compensated  on  the 
average  by  a  horizontal  velocity  of  1597  meters  per  second; 
for  5  kilometers  by  505,  and  for  1  kilometer  by  226  meters  per 
second  by  the  formula, 

(177)  Kinetic  Energy  =  -  \  (q\  -  <?20)  =  [-  ?l  ~  P°1  - 

Pio      —  'i 

=  t-  (ft  - 


The  interesting  fact  appears  to  be  that  each  element  acquires 
the  same  velocity  of  transportation,  so  as  to  make  mixed  masses 
move  together  at  the  same  speed. 

The  Circulation  in  a  Solar  Vortex  or  Sun-Spot 
With  this  preliminary  explanation,  we  can  proceed  to  illus- 
trate the  sun-spot  vortex  by  diagram  and  by  computations. 
Fig.  29  contains  the  general  scheme  of  a  solar  vortex.  Let  the 
normal  1000°—  degree  temperature  levels  be  drawn  for  different 
gases  in  the  adiabatic,  isothermal,  and  the  non-adiabatic  strata. 


258  A  TREATISE   ON  THE   SUN'S  RADIATION 

Let  vertical  convection  raise  a  section  so  that  lateral  horizontal 
motion  takes  place  away  from  a  central  axis  in  all  directions. 
This  drags  after  it  a  simple  funnel-shaped  vortex.  At  the 
extremities  of  horizontal  vortex  motion  the  ends  turn  up  and 
produce  vertical  filamentary  striae,  which  form  a  ring  of  faculae 
at  considerable  distances  from  the  axis.  The  vortex  motion 
lowers  the  isobars  over  the  center,  and  into  this  depression  the 
horizontal  strata  of  the  upper  non-adiabatic  region  descend  in 
slow  inward  spirals,  the  filaments  forming  the  penumbra.  They 
terminate  at  certain  distances  from  the  axis,  according  as  the 
cold  high-level  gases  become  sufficiently  heated  to  lose  their 
visibility.  Over  the  center  of  the  vortex,  called  the  umbra, 
the  gases  may  be  drawn  down  from  great  heights,  affecting  the 
reversing  layer,  the  chromosphere,  and  even  the  hydrogen  of 
the  inner  corona.  These  displacements  all  respond,  in  succes- 
sion, to  the  primary  vortex  under  the  isothermal  layer  which  is 
never  seen,  where  the  powerful  velocities  are  produced  which 
generate  magnetic  field  in  the  core  by  the  rotation  of  the  ions. 
These  are  freely  disintegrated  from  the  atoms  in  the  processes 
of  radiation,  so  that  the  supply  is  as  continuous  as  the  original 
convection.  The  upper  visible  vortex  is  secondary,  and  without 
vortex  motion  proper,  so  that  students  have  been  misled  by  this 

^  fact  in  their  interpretation  of  sun-spots.  A  resurrie  of  the 
facts  given  above  shows  how  the  several  theories  are  related 

\^v>  to  the  true  solar  vortex.  Professor  Hale's  preliminary  descrip- 
tion of  the  formation  of  solar  vortices  is  in  harmony  with  the 
above  account,  though  it  lacks  the  fundamental  isothermal 
action.  The  spectrum  shows  a  series  of  line  and  band  changes 
which  can  readily  be  understood  by  applying  these  principles. 
Indeed,  a  very  complete  solution  of  the  distribution  of  P.  p  .  R.  T. 
at  all  levels  can  be  drawn  out  from  the  Formulas  of  my  Treatise, 
pages  172,  174,  199-202.  We  shall  adopt  these  formulas,  and 
compute  a  few  vortex  tubes,  in  order  to  bring  out  the  dimen- 
sions of  the  vortices  on  the  sun  and  their  attendant  phenomena. 
Very  small  spots  have  a  radius  of  the  umbra  varying  from 
400  to  800  kilometers,  while  in  very  large  spots  the  umbra  has 
a  radius  of  40000  to  50000  kilometers,  the  radius  of  the  pe- 


OTHER   SOLAR  PHENOMENA 


259 


numbra  being  150000  kilometers.  For  a  common  spot  the  out- 
side radii  may  be  taken, 

for  the  umbra,          10000  kilometers, 

"     "    penumbra,    25000 

"     "    faculae,        500000 

We  shall  compute  tubes  between  60474  kilometers  (l)  and 
956  kilometers  (6),  on  a  level  500  kilometers  below  the  reference 
plane,  and  extending  downward  to  50000  kilometers.  Within 
these  adopted  data  nearly  all  sun-spots  will  be  readily  contained. 
Th.e  horizontal  velocities  on  the  500  kilometer  plane  will  range 
from  0.212  to  13.39  kilometers  per  second.  To  find  log  p,  we  take, 
i  (log  60474  -  log  956.25)=  \  (4.78157-2.98057)=  +  0.36020. 
By  a  little  trial  computing,  it  results  that  the  current  function 

$  =  v  w  =  6  400  000  (6.80618),  and  the  tube  constant,  C  =  —  = 

0.0000035  (—  6.54407).  Then  the  successive  values  of  w,  C,  u, 
v,  w,  i,  17,  are  computed  on  the.  500  kilometer  level,  by  the  for- 
mulas indicated.  They  are  next  extended  down  to  the  50000 
kilometer  level.  An  examination  of  the  v,  or  tangential  velo- 
cities, makes  it  probable  that  the  actual  vortex  does  not  extend 
much  beyond  the  horizontal  lines  where  the  velocity  is  2500 
kilometers  per  second.  It  is  likely  that,  with  sufficient  study, 
the  velocities  indicated  by  the  spectroscope  may  be  made  to 
determine  the  theoretical  size  and  power  of  a  given  vortex. 

TABLE  89 
TYPICAL  SOLAR  VORTEX  CONFORMS  TO  KALE'S  DATA 


Radii  in  kilometers,  w2  =  •     -.     Log 


6.80618 


Lines 
LogC 
Kilometers 

(1) 
-6.54407 
Outside 

(2) 
-5.26447 

(3) 
-5.98487 

(4) 
-4.70527 

(5) 
-3.42567 

(6) 
-2.14607 
Center 

21ogp 
+0.72040 

z=            0  
500  

1000  
2000  
5000  .  . 

CO 
60474. 
42762. 
30232. 
19124. 

CO 
26287. 
18658. 
13193. 
8344. 

00 
11513. 
8140.7 
5756.3 
3640.6 

CO 

5023.1 
3551.9 
2511.5 
1588.5 

CO 

2191.7 
1549.8 
1095.8 
693.06 

oo 
956.25 
676.17 
478.12 
302.39 

10000:  

20000  
30000  
40000  
50000  

13523. 
9562. 
7807. 
6761. 
6047. 

5900.1 
4172.0 
3406.5 
2950.1 
2638.6 

2574.3 
1820.3 
1486.3 
1287.2 
1151.2 

1123.2 
794  .  22 
648.49 
561.60 
502.31 

490.08 
346  .  53 
282.94 
245.04 
219.17 

213.82 
151.20 
123.45 
106.91 
95.63 

260 


A  TREATISE   ON  THE   SUN'S  RADIATION 


TABLE  89— Continued 
Radial  velocity  in  kilometers/second,  u  =  Cw 


z  =     0  
500  
1000 

00 

0.2117 
0  1497 

00 

0.4851 
0  3430 

00 

1.1119 
0  7862 

00 

2.5482 
1  8019 

00 

5.8404 
4  1298 

00 

13.3860 
9  4652 

2000  

0  1058 

0  2426 

0  5559 

1  2741 

2  9202 

6  6929 

5000  
10000.  .  .  . 

0.0669 
0  0473 

0.1534 
0  1085 

0.3516 
0  2486 

0.8062 
0  5698 

1  .  8469 
1  3060 

4.2329 
2  9931 

20000  
30000  .  . 

0.0335 
0  0273 

0.0767 
0  0626 

0.1758 
0  1435 

0.4029 
0  3290 

0.9234 
0  7540 

2.1164 
1  7281 

40000 

0  0237 

0  0542 

0  1243 

0  2849 

0  6530 

1  4696 

50000  

0  0212 

0  0485 

0  1112 

0  2548 

0  5840 

1  3386 

Tangential  velocity  in  kilometers/second,  v  =  — 


2  =      0  

500  
1000  

0 
105.83 
149  66 

0 
242.56 
343  02 

0 
555.91 
785  02 

0 
1274.1 
1801  9 

0 

2920.2 
4129  7 

0 
6692.8 
9465  0 

2000  .  . 

211  66 

485  11 

1111  8 

2548  2 

5840  4 

13386 

5000  

334  66 

767  02 

1758  0 

4029  1 

9234  4 

21164. 

10000.  .  .  . 

473  28 

1084  7 

2486  1 

5698  0 

13059 

29931 

20000  

669.33 

1534.0 

3515  8 

8057  8 

18468. 

42327. 

30000  .  .  . 

819  75 

1878  8 

4306  1 

9869  2 

22619 

51842 

40000  
50000 

946.56 
1058  30 

2169.4 
2425  5 

4972.2 
5559  1 

11396. 
12741 

26119. 
29202 

59861. 
66928 

Vertical  velocity  in  kilometers/second,  w  =  —  2  Cz 


z  =>     0.  . 

o 

o 

o 

o 

o 

0 

500  

0.0035 

0.0184 

0.0966 

0.5073 

2.6648 

13.998 

1000  .  . 

0  0070 

0  0368 

0  1932 

1  0146 

5  3211 

27  996 

2000  

0.0140 

0.0735 

0  3863 

2  0292 

10  659 

55.994 

5000.  . 
10000  
20000  
30000  

0.0350 
0.0700 
0.1400 
0  2100 

0.1839 
0.3677 
0.7354 
1  1031 

0.9658 
1.9312 
3.8631 
5  7946 

5.0731 
10.146 
20.292 
30  439 

26.648 
53  .  296 
106.59 
159  89 

139.98 
279.96 
559.93 
839  88 

40000  
50000 

0.2800 
0  3500 

1.4708 
1  8385 

7.7262 
9  6576 

40.585 
50  731 

213.19 
266  48 

1119.8 
1399  8 

Tangential  angle,  tan  i  —  — 


,  _            o 

90°  0'   0" 

90°  0'   0" 

500  
1000  .  . 
2000..      .. 
5000. 

0    6  53 
0   3  27 
0    1  43 
0   0  41 

0   6  53 
0   3  27 
0   1  43 
0   0  41 

10000..      .. 
20000.  . 
30000..      .. 
40000..      .. 
50000..      .. 

0   0  21 
0   0  10 
007 
005 
004 

;• 

0   0  21 
0   0  10 
007 
005 
004 

Vertical  angle,  tan  17  =  

z  =         o  

500»  
1000  
2000  

0°0'    0" 
007 
0   0   10 
0    0    14 
0    0   22 
0    0   31 
0    0   43 
0   0   53 
Oil 
018 

0°  0'    0" 
0   0    16 
0    0   22 
0    0   31 
0    0  49 
0    1    10 
0    1    39 
021 
0   2   20 
0   2  36 

0°  0'     0" 
0    0   36 
0    0   51 
0    1    12 
0    1    53 
0    2   40 
0    3   46 
0    4   37 
056 
0    5    58 

0°  0'     0" 
0     1   22 
0    1    56 
0    2   44 
0    4   20 
067 
0    8   39 
0  10   36 
0  12    15 
0  14      1 

0°  0'    0" 
038 
0    4   26 
0     6   16 
0     9    55 
0  14     2 
0  19    50 
0  24   18 
0  28     3 
0  31   22 

0°    0'    0" 
0      7   11 
0    10    10 
0    14   23 
0    22   44 
0    32    10 
0    45   28 
0    55   42 
1      4    19 
1    11    53 
i 

5000  
10000    .... 

20000 

30000.  . 

40000 

50000  

Formulas.     Meteorological  Treatise,  p.  172 


OTHER  SOLAR  PHENOMENA 


261 


Inward  and  Outward  Velocities  in  Solar  Vortices 

The  velocities  of  inward  and  outward  motions  in  the  solar 
vortex,  made  at  the  Mt.  Wilson  observatory  by  C.  E.  St.  John, 
and  summarized  in  the  Annual  Report  for  1913,  are  interesting 
in  this  connection.  Generally  the  light  gases  have  an  inward 
radial  velocity  and  the  heavy  gases  an  outward  velocity. 

TABLE  90 
SUMMARY  OF  THE  INWARD  AND  OUTWARD  RADIAL  VELOCITIES 


Element 

At.  Wt. 

Line 

Height 
Kilometers 

Vleters/sec. 

Direction 

Hydrogen  .  . 

2 

Ha 

20000 

1500 

Inward 

Calcium  

40 

H7 

H<? 

13000 
4000 
25000 

1000 
200 
2000 

« 
ii 

Magnesium  

24 

H2K2 

4277 
bi  &2 

15000 
3800 
8000 

1300 
100 
400 

41 
II 

Sodium  

23 

(10) 
Di  D2 

3400 
5000 

00 
300 

II 
II 

Aluminum    . 

26 

(20) 

3500 

II 

55 

(10) 

1500 

200 

II 

Lead      . 

207 

(8) 
(7) 
(6) 
(5) 
(4) 
(3) 
(2) 
(1) 
(0) 
(00) 
(000) 

1300 
1100 
1000 
800 
600 
400 
280 
250 
240 
230 
210 
230 

300 
400 
500 
600 
700 
700 
800 
900 
900 
1000 
1200 
900 

Outward 

ii 
ii 

« 
ii 
ii 

< 
i 

« 
i 

Lanthanum  
Barium 

139 
137 

220 
210 

900 
1100 

« 

14 

Niobium  

96 

200 

1200 

|| 

The  same  element,  Hydrogen,  Calcium,  Magnesium,  has 
velocities  decreasing  downward  above,  but  increasing  velocities 
below  the  plane  of  velocity  inversion.  The  heights  assigned 
to  the  inversion  velocities,  about  3500  kilometers  above  the 
photosphere,  should  be  modified  to  variable  depths  below  the 


262  A   TREATISE   ON  THE   SUNJS   RADIATION 

plane  of  reference.  An  adjustment  in  this  respect  is  readily 
interpreted  in  terms  of  the  displacements  of  the  spectrum  lines. 
The  maximum  radial  velocities  observed  seem  to  be  about 
1500  meters  per  second.  On  comparing  such  very  large  radial 
velocities  u  with  those  computed  in  the  typical  vortex  tubes,  it 
is  seen  what  enormous  velocities,  if  those  are  correct  as  ob- 
served, are  implied  in  the  interior  of  the  sun  near  the  axis  of 
the  vortex.  This  subject  requires  a  prolonged  examination. 

The  Invisible  Deep-Seated  Thermodynamic  Processes 
We  have  mentioned  three  processes  that  take  place  near 
the  bottom  of  the  isothermal  layer  during  the  transition  be- 
tween adiabatic  and  isothermal  conditions.  These  are  (l)  the 
generation  of  the  general  radiation,  which  is  afterward  de- 
pleted by  a  small  amount  of  true  absorption  and  a  large  amount 
of  scattering  before  being  reduced  to  the  effective  solar  radia- 
tion toward  the  earth;  (2)  the  production  of  abundant  free 
electric  charges,  afterward  apparent  in  the  sun's  electrostatic 
surface  charge,  in  the  local  magnetic  fields,  and  in  their  varia- 
tions-; (3)  in  the  generation  of  true  vortices  in  certain  latitudes 
and  longitudes.  It  should  be  remarked  that  not  all  of  these 
effects  can  be  readily  observed  from  the  earth,  and^that  there 
are  many  partially  developed  vortices  which  have  only  surface 
changes  in  the  areas  of  the  faculae,  being  the  symptoms  of  deeper 
seated  operations.  Consequently,  there  must  be  many  maxima 
and  minima  in  the  output  of  the  solar  radiation,  which  are  only 
imperfectly  registered  in  the  high  level  phenomena  above  the 
photosphere.  If  the  spectroheliograph  has  such  difficulty  in 
fixing  accurately  the  levels  and  the  velocities  of  the  movements 
of  large  masses  of  gas  and  vapor,  it  is  evident  that  we  need  not 
feel  constrained  to  limit  our  information  to  its  particular  evi- 
dences. If  there  are  registered  at  the  earth  successive  maxima 
and  minima  along  the  solar  surface  in  its  synodic  rotation,  as 
changes  in  the  intensity  of  radiation,  in  the  magnetic  field,  in 
the  temperatures  and  pressures  of  the  earth's  atmosphere,  there 
is  no  good  reason  derived  from  spectrum  observations,  whether 
visual  or  thermal,  why  such  disturbances  may  not  occur  be- 


OTHER  SOLAR  PHENOMENA  263 

neath  the  isothermal  layer,  even  if  there  are  no  spots  and  no 
important  groups  of  faculae  above  that  layer.  It  is  known  that 
magnetic  storms  originate  at  the  earth  from  solar  impulses, 
without  attendant  sun-spots,  and  there  can  be  no  reason  why 
this  and  similar  conditions  may  not  occur  under  the  invisible 
solar  processes. 

The  General  Circulation  in  the  Sun  in  Latitude 

From  a  study  of  the  effects  of  the  circulation  from  within 
the  sun,  as  shown  in  several  surface  phenomena,  a  suggestion 
may  be  ventured  regarding  the  thermal  movements  below  the 
isothermal  layer.  The  statistical  studies  of  the  relative  fre- 
quency of  the  sun-spots,  the  faculae  and  the  prominences,  to- 
gether with  the  extensions  of  the  solar  corona,  will  serve  to 
indicate  the  zones  of  maximum  vertical  convection  in  the  lati- 
tude, that  is,  along  the  meridians;  also  the  maxima  and  minima 
of  the  solar  convection  in  longitude,  along  planes  parallel  to  the 
equator,  are  registered  in  the  variations  of  the  terrestrial  mag- 
netic field. 

The  relative  frequencies  of  the  solar  prominences,  faculae, 
and  sun-spots,  as  observed  by  the  Italian  spectroscopists, 
Secchi,  Tacchini,  Ricco,  and  others;  by  Wolf,  Wolfer,  and  others, 
have  been  studied  in  10-degree  zones,  in  a  series  of  papers  by 
Lockeyer,  1902,  1903,  1904,  Proc.  Roy.  Soc.;  by  Bigelow, 
Monthly  Weather  Review,  Jan.,  1903;  by  Ricco,  Mem.  Spett. 
Ital.,  1903,  1914.  These  papers  reach  similar  conclusions  re- 
garding the  movements  in  latitude,  and  for  a  summary  the 
data  of  my  paper  are  quoted  covering  the  mean  values,  1872- 
1900.  Three  11-year  cycles  are  superposed,  and  the  mean 
values  are  given  in  Table  91  for  each  10-degree  zone  north  and 
south  of  the  equator  during  a  common  period  of  1 1  years.  These 
data  were  plotted  in  Fig.  3  of  my  paper,  and  there  appear 
to  be  maxima  which  have  a  movement  in  latitude  as  well  as 
in  intensity.  Scale  from  the  diagram  the  maximum  ordinates 
for  successive  years,  and  plot  them  as  on  Fig.  30,  making  at 
the  given  latitudes  a  series  of  points  measured  from  the  limb, 
which  are  proportional  to  these  frequencies.  The  maxima  have 


264  A   TREATISE   ON  THE   SUN'S   RADIATION 


Circulation  along  the          \ 
Solar  Meridians 


II  50 


Circulation  parallel  to 
the  Solar  Equator 

FIG.  30.     Circulation  In  the  Interior  of  the  Sun. 


OTHER  SOLAR  PHENOMENA  265 

two  branches  for  the  prominences  in  each  hemisphere,  (1)  form- 
ing a  narrow  loop  near  latitudes  =*=  20°,  and  (2)  forming  a  broad 
loop  between  latitudes  =*=  50°  and  =±=  70°.  The  figures  show 
that  the  maxima  progress  in  two  cycles,  first,  toward  the  equator, 
secondly,  toward  the  poles.  The  solar  impulse  is  apparently 
outward  near  latitude  =*=  40°,  in  zones  surrounding  the  sun, 
with  quasi-vortical  curls  on  each  side  of  the  principal  axes  or 
planes.  The  curl- vortex  circulation  is  common  to  nearly  all 
free  convectional  movements,  involving  a  central  upward  con- 
vection, and  two  lateral  downward  branches  of  descending 
materials.  If  these  ascending  and  descending  branches  form 
parts  of  a  convectional  circulation,  there  will  evidently  be  pro- 
duced a  series  of  oval-elliptical  stream  lines,  narrow  near  the 
interior  and  broad  near  the  surface  of  the  solar  mass.  There 
seem  to  be  two  complete  circuits  in  each  hemisphere,  and  the 
total  forms  a  sort  of  zonal  harmonic  with  four  axes,  taking  the 
sun  as  a  whole.  There  are  many  details  which  harmonize  with 
this  general  view.  The  maxima  of  the  sun-spot  and  the  faculae 
frequencies  form  circuits  from  the  latitudes  =*=  30°  toward  the 
equator.  This  is  easily  seen  to  be  the  effect  which  would  follow 
if  a  renewal  of  the  general  convection,  during  the  sharply  ascend- 
ing branch  of  the  11 -year  cycle,  should  cause  the  throat  of  the 
ascending  convection,  with  its  lateral  branches,  to  reach  higher 
up  toward  the  surface,  so  that  during  the  first  years  of  the 
period  the  maximum  outbursts  are  nearer  together,  =*=  50°  to 
=*=  30°  in  latitude;  this  vortex  may  then  fall  back  toward  the 
center  of  the  sun,  and  cause  the  maximum  points  to  recede 
toward  the  equator  and  the  poles,  respectively.  This  general 
zonal  movement  in  latitude  conforms  with  the  observed  fre- 
quency of  these  surface  phenomena  in  latitude,  and  refers  them 
to  the  general  thermodynamic  convection  within  the  great  in- 
terior masses  of  the  sun.  They  are  leisurely  and  persistent, 
and  they  are  produced  and  sustained  by  the  pressure  of  gravi- 
tation, which  causes  hydrostatic  reactions,  circulation,  and  the 
generation  of  free  heat  as  one  of  the  terms  in  the  formation  of 
the  general  solar  radiation. 


266  A   TREATISE   ON  THE   SUN'S  RADIATION 

DOOCOOOiO^OOOGOCOOC 
TSrHCDrHrHO'O^O'OO-IC^I 


Is 

1    I 


Is 

1    1 


00 

<o  t~ 

1  1 


83 

1    I 


1     I 


00 
I 


00 


00 

CON 


00 


t^ 


(NI>»O'^COO 
CO  i-H  T-H  1-1 


COOOOOi—  irJHiOCOCOCOO5(N 
t>»CO         T—  ii—  II-HI—  1         i—  (  00  Oi 


1>  OS.t^  rH  i—  1  (M  C<l 
OOOiOI>»OCOOi 


i-H   C<1    i—  1   f-H    i—  1 


COiO(MI^^OO(MCOQ 
OiCOOGOCOlMi-HCOO 


lOrHOCDC5CiCOrH<MrHrH 


COCOO5rHOOOO<MO<NCOrH 
rHTtHrHOSOOt-^OOI^rHO 


<N    <M    r- 1    r- 1    rH    TH  rH    C5    O 


1>1^-«*IO>OO»OCDCO»OOOOO 

rH  rH    rH  rH    (M 


lOC^COlOOiCOCOC^C^COrH 


0°      I        rHrH 

^Z  -44  I  ^T1      T~H 


^rHCOOOOrHrHrHOOlNOO 


OOOrHrHrHOOlN 


00 

m  TJI 
1    1 


1    1 


s§ 

I  I 


00 


TH  O^  t^  ^D 

rH   (^ 


00 


00 


rH   rH  rH   rH    1C   O 


'*'*OO  OOCO 


iOl>(NCOCQ 
CO  CO   W   rH   rH 


;7  i 


I     rH      |        |        |        |        |     10  CD   CO 


I>COCO(M      1        iT^rHCOOO 
<N   <N   rH  1        1  (M   CO  l>  CD 


ss 

1  1 


00 

7 


O^oO 
COOOO 


IO 


Tt^Ot^O5l>iO(NCOiOCOOO 

<£>GOCOrHrH  1-HCDO5O5 


T^lOCOrHrH      I     (MOrHCOl> 

I        c<i  •*  •<*  co 


00      I 


|        |        |         |        | 


OTHER  SOLAR  PHENOMENA  267 

The  General  Circulation  of  the  Sun  in  Longitude 

In  Bulletin  No.  21,  U.  S.  Weather  Bureau,  the  results  were 
published  of  a  study  of  the  relative  intensity  of  the  variations 
of  the  terrestrial  magnetic  field  in  the  adopted  equatorial 
synodic  period  of  26.68  days.  Compare  Chapter  5  and  Fig.  23 ; 
also  Meteorological  Treatise,  pp.  329-335.  These  statistical 
studies  of  the  relative  maxima  and  minima  were  suggestive  of 
effects  produced  by  the  solar  electro-magnetic  radiation  upon 
the  terrestrial  magnetic  field.  They  were  persistent  in  longi- 
tude, the  compilations*  covering  the  interval  1842-1896;  the 
series  has  been  extended  to  1915.  There  are,  of  course,  many 
irregularities  to  consider,  but  generally  it  seems  as  if  the  solar 
output  is  very  persistent  as  to  its  maxima  in  certain  longitudes. 
At  the  time  of  the  original  research,  1892-95,  before  the  dis- 
covery of  the  process  of  the  ionization  of  rarefied  gases  under 
the  impact  of  the  solar  waves  of  high  frequency,  it  was  difficult 
to  assign  a  physical  process  competent  to  produce  such  mag- 
netic variations.  The  statistical  efforts  to  connect  sun-spots 
with  the  magnetic  disturbances  are  only  in  part  successful,  but 
it  is  easy  to  see  that  many  maxima  impulses  may  occur  in  radia- 
tion when  the  isothermal  layer  has  not  been  broken  through 
in  the  sun-spot  vortical  circulation.  The  radiation  maxima 
may  be  nearer  the  equator  than  that  of  the  sun-spot  develop- 
ment in  higher  latitudes.  The  great  tidal  wave  of  the  solar 
circulation  sweeps  with  its  maximum  angular  velocity  along 
the  sun's  equator  in  26.68  days,  but  this  trails  off  in  higher  lati- 
tudes to  more  than  30.00  days  near  the  poles.  It  is,  however,  the 
26.68-day  period  which  is  dominant  in  the  terrestrial  magnetic 
field. 

We  have  reproduced  on  Fig.  30  the  periodic  curve  of  varia- 
tion, with  its  eight  maxima  and  minima  in  Longitude,  I,  II, 
.  .  .  VII,  VIII.  It  is  seen  that  they 'are  quite  uniformly  dis- 
tributed around  the  axis  of  rotation,  and  we  may  suppose  that 
there  are  eight  principal  vertical  axes  of  vortical  convection 
along  and  near  the  solar  equator.  Admitting  the  vertical  cur- 
rent with  lateral  descending  branches,  we  have  again  a  circiila- 


268 

tion  in  eight  branches  related  as  vortical  curls.  It  does  not 
seem  to  be  unnatural  that  the  convectional  movements  in  the 
sun  should  sustain  two  zones  in  latitude,  one  in  each  hemisphere, 
and  that  these  should  concentrate  in  maxima  and  minima  in 
longitudes,  according  to  their  inherent  thermodynamic  processes. 
If  this  is  the  natural  condition  of  large  bodies  of  gaseous  ma- 
terials, cooling  under  gravitation,  it  then  becomes  proper  to 
admit  that  certain  longitudes  and  latitudes  accumulate  maxi- 
mum conditions  of  radiation.  It  has  not  seemed  to  me  pos- 
sible that  such  magnetic  variations  can  persist  so  steadily  with- 
out some  corresponding  constitution  in  "the  solar  mass.  What 
the  harmonic  laws  of  the  thermodynamic  process  are  in  them- 
selves we  are  not  in  position  to  explain,  but  the  preceding  analy- 
sis may  provide  some  valuable  suggestions  in  this  connection. 

The  Chromosphere  and  the  Inner  Corona 

The  gaseous  distribution  of  the  solar  elements  has  given 
rise  to  three  popular  names  for  their  visible  appearance:  (l)  The 
photosphere,  or  sharp  limb;  (2)  the  chromosphere,  or  thin 
luminous  ring  just  above  the  photosphere;  (3)  the  inner  corona, 
or  apparent  limit  of  a  gaseous  atmosphere.  These  divisions 
have  a  general  connection  with  the  vanishing  heights  of  the 
heavy  and  light  gases,  as  indicated  on  Table  8  and  Fig.  18. 
For  the  temperature  T  =  0°  of  the  several  gases,  the  hyperbolic 
law  for  the  vanishing  height  is, 

mzT  =  0  =  45400  kilometers 

Since  Calcium,  m  =  40,  just  floats  conspicuously  on  or  near  the 
flash-spectrum  stratum,  z  =  400  —  500,  this  may  be  taken  as 
the  limit  of  the  heavy  photospheric  layers;  Carbon,  m  =  12, 
may  be  taken  as  the  limit  of  the  true  chromosphere;  hydrogen, 
m  =  2,  is  the  limit  of  the  inner  corona. 

Inner  Corona,    H  =    2,  Height  =  23000  to  3800  kilometers 
Chromosphere,  C  =  12,       "      =    3800  to  1100 
Photosphere,    Ca  =  40,       "      =    1100  to    200 


OTHER  SOLAR  PHENOMENA  269 

These  merge  into  one  another  as  the  gases  vanish  at  heights 
corresponding  with  their  atomic  weights.  In  violent  convec- 
tions and  eruptions  there  are  temporary  intrusions  of  the  heavy 
gases  above  their  normal  vanishing  levels,  but  gravitation  seeks 
to  restore  them  to  those  levels  by  means  of  the  thermodynamic 
processes  that  have  been  described.  The  heights  that  have 
been  assigned  by  other  writers  to  the  gases  in  the  chromosphere 
were  derived  generally  from  a  study  of  the  relative  length  of 
the  arcs  as  photographed  in  the  several  types  of  spectra.  Many 
of  these  results  are  inconsistent  with  the  thermodynamic  data. 
The  spectrum  lines  depend  upon  so  many  factors  besides  den- 
sity and  height  that  it  will  be  proper  to  revise  this  subject  in 
connection  with  the  data  that  have  been  obtained  in  this  com- 
putation. Compare  Abbot's  "Sun,"  pages  137-182,  for  details 
and  general  information. 

The  Outer  Solar  Corona 

The  true  gaseous  atmosphere  of  the  sun  is  surrounded  by  a 
very  extended  and  excessively  tenuous  appendage  called  the 
outer  corona,  which  is  visible  during  the  minutes  of  total  eclipses 
of  the  sun.  It  is  closely  associated  with  the  underlying  gases 
and  their  thermodynamic  processes.  Generally,  it  has  a  quad- 
rilateral shape  with  maximum  wings  near  latitudes  =•=  40°, 
just  over  the  axes  of  the  vertical  convection,  I,  II,  III,  IV, 
on  Fig.  30.  At  the  minimum  of  the  11-year  period  the  exten- 
sions are  mainly  equatorial;  they  are  not  large  over  the  ascend- 
ing branch  of  the  deep- throated  vortex;  while  the  polar  regions 
are  covered  with  short,  individual  rays,  which  conform  in  shape 
closely  to  the  lines  of  force  that  would  surround  the  sun  if  it 
were  in  fact  a  large  spherical  magnet.  The  quadrilateral  forms 
occur  during  the  rise  and  fall  of  the  relative  intensity  of  the 
solar  convection;  at  the  maximum  the  sun  is  surrounded  by 
coronal  streamers  of  fantastic  shape  and  irregular  structure. 
This  change  of  form  of  the  corona  is  closely  connected  with  the 
11-year  cyclic  changes  in  the  frequency  of  the  sun-spots,  faculae, 
prominences,  the  intensity  of  the  solar  radiation  into  space,  the 


270  A  TREATISE   ON  THE   SUN'S   RADIATION 

variations  in  the  terrestrial  magnetic  field,  precipitation,  tem- 
perature, vapor  pressure,  barometric  pressure,  and  common  cli- 
matic conditions.  As  these  have  been  described  in  the  Meteor- 
ological Treatise,  the  reader  can  be  referred  to  that  place  for 
further  data.  The  subject  of  solar  and  terrestrial  synchronism 
in  the  11-year,  3.75-year,  and  shorter  cycles  has  been  so  ex- 
tensively studied  and  verified  in  all  parts  of  the  world  during 
long  intervals  of  time  that  it  is  accepted  as  a  general  phenomena 
in  solar  physics.  While  further  researches  are  required  to  de- 
termine its  processes  in  detail,  the  consensus  of  opinion  is  grow- 
ing that  it  is  a  subject  of  great  economic  value,  and  that, 
although  complex,  there  is  every  prospect  of  complete  solutions 
of  the  principal  phenomena. 

The  distribution  of  the  intensity  of  the  coronal  radiation 
has  been  the  subject  of  many  researches  in  eclipse  expeditions. 
There  is  distinct  radial  polarization,  vanishing  at  the  photo- 
sphere and  increasing  outward,  as  if  due  to  the  scattering  of 
reflected  light  on  isolated  particles  of  matter  thrown  outward 
under  the  pressure  of  radiation.  These  minute  particles  of 
matter,  produced  in  the  general  processes  of  thermodynamics, 
pervade  all  the  gaseous  strata,  as  well  as  the  corona,  in  propor- 
tion to  the  local  forces  in  action,  more  thickly  below  than  above. 
Hence,  polarized  reflection  above  becomes  diffused*  scattering 
below  in  conformity  with  the  observations.  The  bright  coronal 
line  in  the  spectrum  5303  is  assigned  to  an  unknown  element, 
"  coronium,"  and  there  are  several  other  bright  lines.  The 
Fraunhofer  dark  lines  of  the  photospheric  spectrum  have  been 
observed,  and  this  is  evidence  of  reflected  light  coming  from  the 
lower  levels.  The  incandescence  of  the  coronal  matter  is  ap- 
parently a  feeble  effect,  taken  by  itself,  and  the  luminous  appen- 
dage is  chiefly  dependent  upon  minute  particles  of  matter  in 
the  neighborhood  of  the  sun,  which  reflect  and  scatter  the  true 
radiation  passing  through  them.  These  small  particles  are  prob- 
ably charged  with  electricity,  either  from  local  ionization  or 
from  electric  charges  which  are  transported  from  the  isothermal 
strata.  There  is  evidence  of  the  structural  arrangement  of 
these  particles  along  the  magnetic  lines  of  force  in  the  polar 


OTHER  SOLAR  PHENOMENA  271 

regions,  and  along  the  electrostatic  lines  of  repulsion  in  the 
equatorial  wings,  extending  to  many  diameters  of  the  sun  in 
all  directions  near  that  plane.  Small  particles  are  transported 
by  the  pressure  of  light  after  they  have  been  detached,  just 
as  comet-tails  are  formed  by  surface  heating  from  solar  radia- 
tion, which  sets  free  the  matter  that  is  then  carried  into  space 
on  the  waves  of  light.  The  corona  presents  many  confused 
phenomena  which  interact  upon  one  another. 

Solar  Magnetism 

The  general  view  tha.t  the  sun  is  a  magnetic  sphere  has  been 
a  subject  of  discussion  for  a  century.  There  has  been  unques- 
tioned evidence  that  the  variations  of  the  terrestrial  magnetic 
field  follow  those  of  the  sun-spot  and  prominence  frequencies 
in  a  persistent  synchronism.  That  there  is  some  causal  connec- 
tion between  the  entire  solar  and  terrestrial  fields  is  beyond 
debate,  although  the  difficulty  of  understanding  the  natural 
mechanism  has  not  even  yet  been  fully  overcome.  The  theory 
that  the  sun  is  a  magnetic  sphere,  competent  to  affect  the  ter- 
restrial magnetic  field  by  direct  action  to  the  extent  that  is 
observed,  has  always  been  opposed  by  two  arguments:  (1)  The 
high  temperature  of  the  sun  precludes  its  retention  of  internal 
magnetism;  (2)  the  great  distance  between  the  two  bodies 
would  require  an  excessive  magnetization  in  the  interior  of  the 
sun.  Two  equally  obstinate  facts  have  stood  against  these 
arguments:  (1)  The  earth  retains  a  permanent  magnetic  field 
although  its  interior  is  at  a  very  high  temperature;  (2)  the  syn- 
chronous connection  can  not  be  denied,  and,  indeed,  its  economic 
importance  in  connection  with  long-period  forecasts  of  weather 
conditions  is  so  great  that  the  subject  justifies  research  till  the 
physical  laws  become  well  understood.  My  own  researches  in 
the  years  1889-1891  afforded  me  sufficient  reason  for  proceeding 
steadily  in  this  general  problem  without  the  least  hesitation, 
the  difficulties  being  an  incident  in  the  progress  of  science. 

The  entire  subject  has  been  greatly  illuminated  by  the  prog- 
ress of  physics  in  the  direction  of  atomic  ionization,  which 


272 

showed  two  facts,  first,  that  the  necessary  solar  impulses  can 
be  propagated  to  the  earth  along  the  electromagnetic  field,  thus 
relieving  us  of  the  necessity  of  ascribing  to  the  solar  magnetiza- 
tion an  excessive  intensity;  secondly,  the  identification  of  the 
Zeeman  Effect  in  the  sun-spots,  and  in  other  regions  of  the  sun, 
so  that  the  high- temperature  argument  against  solar  magnetism 
has  fallen  out  of  the  discussion.  The  general  result  is  to  leave 
the  sun  a  low-power  spherical  magnet  and  the  solar  radiation 
as  the  carrier  of  the  energy  which  is  observed  in  the  earth's 
atmosphere  in  many  forms  of  synchronism.  Since  electrons, 
or  free  charges  of  electricity,  in  combination  with  atoms  and 
molecules,  or  entirely  apart  from  them,  exist  in  the  sun's  atmos- 
phere, their  movements  induce  magnetic  field,  under  rotation 
or  during  their  translation,  so  that  general  and  local  magnetic 
fields  are  a  fundamental  part  of  the  sun's  constitution.  The 
credit  for  determining  the  existence  of  the  Zeeman  Effect  in  the 
sun  is  due  to  the  Director  of  the^  Mt.  Wilson  observatory,  Dr. 
George  E.  Hale,  and  the  allied  solar  phenomena  are  receiving 
constant  study  with  the  special  equipment  that  has  been 
acquired  for  the  purpose.  The  phenomenon  of  the  ionization  of 
gases  received  its  greatest  impulse  from  Sir.  J.  J.  Thomson, 
of  the  Cambridge  Physical  Laboratory,  in  1898,  and  it  has 
now  penetrated  the  remotest  branches  of  Physical  Science. 
Dr.  Hale's  discovery  was  published  in  1908,  and  it  is  now  of 
common  astrophysical  interest.  During  the  years  preceding 
1908  the  subject  of  solar  magnetism  was  in  the  debatable  stage, 
but  since  that  year  it  has  been  marked  by  steady  progress. 
The  preceding  data  from  thermodynamic  computations,  as  to 
the  physical  conditions  under  which  the  solar  gases  are  con- 
trolled, afford  much  additional  ground  for  believing  that  the 
solar-terrestrial  synchronism  will  finally  become  a  complete  and 
practical  branch  of  Astrophysical  Meteorology.  A  few  details 
will  now  be  given,  as  a  summary  of  the  most  important  facts 
that  are  known. 


OTHER  SOLAR  PHENOMENA  273 

The  Polar  Rays  of  the  Solar  Coronas  as  Evidence  of  a  Magnetized 

Sphere 

During  the  minimum  years  of  the  solar  11 -year  cycle  the 
polar  rays  stand  apart  with  much  individuality,  so  that  their 
paths  in  polar  coordinates  (r.6)  can  be.  measured  at  several 
points  along  each  ray.  The  coronas  of  February  29,  1878, 
January  1,  1889,  and  December  22,  1889,  were  studied  especially 
to  determine  if  there  is  any  period  of  rotation  common  to  them, 
such  that  one  model  rotated  in  this  period,  and  placed  accord- 
ing to  the  axes  S  of  the  sun's  rotation,  E  of  the  earth's  rotation, 
and  K  of  the  ecliptic,  would  reproduce  the  successive  aspects  of 
the  coronal  field.  The  result  may  be  stated  in  a  few  paragraphs: 

1.  Period  of  sidereal  rotation,  27.4117  days. 

"       "  synodic      "          29.6358    " 

Mean  daily  motion  (sidereal)  in  longitude  13°.  1331.  This 
applies  to  the  pole  of  the  corona  C,  which  does  not  coincide 
with  the  pole  of  the  sun's  rotation. 

2.  Distance  between  the  pole  of  the  sun  and  the  pole  of 
the  corona,  S  C  =  4°  30'  approximately. 

3.  The  south  coronal  pole  is  100°  in  advance  of  the  north 
coronal  pole.    An  ephemeris  may  be  constructed  from  the  fol- 
lowing data: 


Epoch 

Period 

North  Coronal  Pole 

South  Coronal  Pole 

1878.0 

27.4117 

Latitude,       85°  32' 
Longitude,  201  °.2 

Latitude,    -85°  24' 
Longitude,  301  °.6 

The  axis  of  polarization  is  therefore  at  the  surface  of  the 
sun  about  4i°  from  the  axis  of  rotation,  and  the  southern  end 
of  it  precedes  by  about  100  degrees  in  longitude. 

4.  The  rays  of  the  solar  corona  seem  to  originate  in  a  belt 
whose  mean  distance  from  the  poles  of  the  corona  is  about 
32°  or  33°.  The  bases  of  these  polar  rays  indicate  that  there 
is  a  zone  of  maximum  output  of  the  small  particles  which,  by 
the  arrangement  of  them  in  curves  under  the  influence  of  the 
solar  magnetic  lines  of  force,  make  up  the  visible  rays. 


274  A   TREATISE   ON  THE   SUN'S   RADIATION 

5.  The  curvature  and  the  distribution  of  the  group  of  polar 
rays  conform  to  the  lines  of  force  of  a  spherical  magnet  when 
they  are  seen  projected  on  a  plane  which  is  normal  to  the  line 
of  sight. 

(178)  Equation  of  the  lines  of  force,  N  =  •—.      — , 

o  T 

where  TV  is  a  constant  along  one  ray,  but  varies  from  one  ray 
to  another. 

6.  It  will  be  noted,  on  comparing  with  the  diagram  of  the 
prominences,  Fig.  30,  that  the  coronal  belt  coincides  closely  in 
latitude  with  the  position  of  the  maximum  of  the  prominences 
during  the  development  of  the  descending  branch  of  the  11-year 
curves,  that  is,  from  6  to  11  years,  so  that  they  are  closely  as- 
sociated with  the  poleward  branch  of  the  general  vortex  of  cir- 
culation. 

7.  The  synchronism  between  the  solar  and  the  terrestrial 
phenomena  is  based  upon  the  26.68-day  period,  along  the  solar 
equator  so  that  these  are  propagated  along  the  electro-magnetic 
field  of  radiation. 

8.  The  periods  above  give  us  for  the 

Base  of  the  coronal  rays,  29.64  days;  788'  angular  velocity. 
Equatorial  synchronism,  26.68    "       864'       "       .     " 
Comparing  with  Table  81,  "Treatise,"  it  seems  that  at  the 
surface  788'  occurs  in  latitude  45°  as  compared  with  the  spectrum 
displacements.    Since  the  high-level  hydrogen  has  larger  angular 
velocities,  as  830'  daily,  it  is  quite  likely  that  the  base  of  the 
coronal  lines  is  near  the  top  of    the  chromosphere,  at  5000 
kilometers. 

9.  At  the  time  my  research  was  made,  1889-1891,  it  was 
commonly  accepted  that  the  solar  synodic  period  was  from  many 
sources  about  26.00.    The  fact  that  the  terrestrial  magnetic  field 
was  at  that  time  competent  to  yield  a  period  so  closely  conform- 
ing to  recent  spectroscopic  results  (870'  to  880')  is  testimony  to 
the  reality  of  the  solar  impulses  in  the  earth's  atmosphere. 

10.  There  are  reasons  to  think  that  this  transmission  is  due 
to  a  bombardment  of  solar  corpuscles  carried  along  on  the  light- 
wave front;    other  reasons  suggest  that  the  terrestrial  electric 


OTHER   SOLAR  PHENOMENA 


275 


charges  are  due  to  the  disintegration  of  the  atoms  and  mole- 
cules in  the  high,  rarefied  strata  of  the  earth's  atmosphere,  under 
the  impact  of  the  solar  radiation  of  short-wave  lengths.  Since 
there  is  clear  evidence  of  a  large  depletion  of  short  waves  in 
the  earth's  atmosphere,  it  would  seem  that  this  is  the  pre- 
dominant cause  of  the  high-level  ionization. 


The  Zeeman  Effect  in  the  Sun's  Atmosphere 

The  Zeeman  Effect  was  discovered  in  1896,  and  it  consists  in 
splitting  up  a  simple  spectrum  line  into  two  circularly  polarized 
or  three  plane  polarized  lines,  when  the  ray  in  a  strong  magne- 
tic field  is  viewed  along  or  normal 
to  the  lines  of  magnetic  force  H. 

Let  H  =  the  magnetic  intensity 
along  the  axis  z,  at  right  angles  to 
the  plane  x  .  y,  and  let  the  ion  have 
the  mass  m,  charge  e}  and  let  the 

coefficient  of  elasticity  to  the  cen-      z< H  c 

ter    be  k,  then   the   equations   of 
motion  of  e  m  are : 


(17Q\    <m   — —    — £2  ~  _L  p  7,      TJ 

\ J.  I  \J )     fii      j  K     A>      |^  t/  u  •  I~l 

(t  t 

dv      • 

m  -T7  =  —  kzy  —  eu  .  H 
a  t 


FIG.  31.     Elements  of  the  Zee- 
man Effect. 


These  equations  are  solved  by  the  following  terms: 

(180)  x  =  x0  est,  provided,  msz  x0  =  —  k2  x0  +  e  H  .  s  y0, 

(181)  y  =  yQ  est,  provided,  m.  s2  yQ  =  —  k2  yQ  —  e  H  .  s  x0, 

,.  0_x      ,  .    k         .2  IT  ,  ^,27T  Vw  .    .,          .    , 

(182)  where  5  =  ^—F=.  =  ^  —  =,  and  T — - —  is  the  period. 

Vw          T  k 

This  occurs  when  H  =  0,  and  there  is  no  magnetic  field,  so 
that  T  represents  the  undisturbed  periodic  motion  of  the  line- 
constituents.  If  H  has  a  value, 

»      27TVW 


276  A   TREATISE   ON  THE   SUN'S   RADIATION 

The  sign  +  signifies  a  greater  periodic  motion  for  the  posi- 
tive rotation,  and  the  sign  —  for  a  smaller  period. 

(1)  When  viewed  along  the  magnetic  field  the  two  circular 
components  accelerate  or  retard  the  normal  rotation,  change 
the  wave  length  and  the  position  of  two  resultants  in  the  spec- 
trum. The  original  spectral  line  becomes  two  circularly  polar- 
ized lines  of  equal  intensity  rotating  in  opposite  directions,  and 
symmetrically  displaced  in  respect  of  the  undisturbed  position. 
(2)  When  the  ion  is  viewed  across  the  magnetic  field,  at  right 
angles  to  H,  the  component  along  z  is  unaltered  in  period  and 
position;  the  two  circular  components,  seen  in  the  xy  plane, 
appear  to  be  plane  polarized,  the  direction  of  vibration  being  at 
right  angles  to  the  central  s-component.  The  spectral  line  is 
broken  up  into  three  distinct  plane  polarized  lines,  the  central 
vibrating  parallel  to  the  magnetic  field  and  the  two  outer 
components  vibrating  at  right  angles  to  it. 

Since  the  polarization  may  be  circular,  elliptical,  or  com- 
plex, it  follows  that  there  may  be  developed  several  component 
of  these  two  principal  types,  and  numerous  lines  have  been  deter- 
mined by  experiment  in  strong  magnetic  fields.  Such  division 
of  a  simple  spectral  line  into  components  is,  therefore,  proof 
that  a  magnetic  field  exists  of  sufficient  strength  to  be  detected 
and  measured.  This  method  was  successfully  applied  by  Dr. 
G.  E.  Hale,  in  1908,  to  the  sun-spots,  wherein  the  spectrum 
lines  were  broadened  and  subdivided  in  conformity  with  these 
tests.  This  constitutes  proof  that  free  ions  or  electrons  (e.m) 
exist  in  the  sun-spots,  and  that  the  vortical  motion  is  suffi- 
ciently rapid  to  produce  strong  magnetic  fields.  Similarly,  the 
Zeeman  Effect  has  been  generally  found  in  the  sun's  atmos- 
phere, and  its  corresponding  magnetic  field  has  been  deter- 
mined, so  that  the  subject  of  solar  magnetization  of  the  entire 
mass,  or  local  magnetic  fields  accompanied  by  electric  currents, 
becomes  a  very  important  subject  of  research.  The  literature 
is  already  extensive,  so  that  only  a  few  conclusions  can  be 
mentioned.  This  simple  theory  of  accounting  for  the  Zeeman 
Effect  was  first  proposed  by  Lorentz,  but  at  has  been  extended 
to  comprise  very  complex  motions,  including  the  rotation  of 


OTHER  SOLAR  PHENOMENA  277 

the  axes  themselves.  The  ions  in  high  temperature  gases  may 
by  their  motions  produce  the  most  varied  kinds  of  optical  spectral 
effects. 

The  Distribution  of  the  Solar  Magnetism  as  Determined  by  the 

Zeeman  Effect 

If  a  positive  charge  rotates  anti-clockwise  and  a  negative 
charge  rotates  clockwise,  a  positive  magnetic  field  is  generated 
along  the  z-axis.  Hence,  by  means  of  the  Zeeman  Effect,  it 
is  possible  to  determine  the  polarity  in  the  sun-spots  and  in 
other  localities  on  the  sun.  The  observations  on  the  sun- 
spots  show  that  there  are  in  each  hemisphere  about  as  many 
of  one  polarity  as  the  other;  that  as  the  sun-spots  are  com- 
monly generated  in  pairs  they  are  likely  to  be  of  opposite 
polarities;  that  the  intensity  of  the  magnetic  field  in  the  sun- 
spots  increases  radially  toward  the  axis  and  downward  from 
the  plane  of  the  photosphere;  that  the  effect  is  distributed  in 
many  anomalous  ways  among  the  lines  of  the  spectrum.  Dr. 
Hale  has  properly  associated  these  phenomena  with  a  true 
vortex  motion,  wherein  the  inner  tubes  rotate  faster  than  the 
outer  and  more  rapidly  with  the  depth  below  the  reference 
plane.  It  is  thought  that  the  introduction  of  the  isothermal 
shell,  the  depths  of  the  several  gases,  the  computed  P,  p,  R,  T, 
are  all  in  complete  harmony  with  the  requirements  of  the  solar 
spectrum.  If  the  ions  are  chiefly  generated  near  the  bottom 
of  the  isothermal  layer  of  each  gas,  the  intrusion  of  abnormal 
temperatures  from  below,  as  well  as  from  above,  will  induce 
segregation  of  the  ions,  the  negative  going  to  the  cooler  and 
the  positive  to  the  warmer  strata,  in  a  general  way.  Since  the 
deflecting  force  of  rotation  near  the  equator  is  small  on  the 
sun,  it  follows  that  the  underflowing  hot  sheet  will  rotate  indif- 
ferently in  forming  the  dependent  vortex,  so  that  it  is  probable 
that  a  group  of  sun-spots  consists  of  independent  axes  of  funnel- 
shaped  tubes,  rather  than  of  curved  horseshoe  vortices  termi- 
nating on  the  free  surface  of  a  stratum,  since  no  such  definite 
layer  exists  for  the  light  gases,  though  by  complex  interaction 
between  light  and  heavy  gases  some  such  complex  layer  may 


278  A   TREATISE   ON   THE   SUN'S   RADIATION 

be  formed  as  a  photospheric  stratum.  The  structure  of  this 
region  is  so  very  complicated  that  the  tangled  record  of  the 
spectrum  is  to  be  expected.  There  are  many  statements  re- 
garding the  height  of  the  lines  of  flow  which  will  need  to  be 
modified.  The  method  of  double  reversal,  or  that  of  the  long 
and  short  arcs,  while  they  give  indications  of  the  height,  should 
be  interpreted  in  terms  of  the  general  thermodynamic  require- 
ments that  must  prevail  on  the  sun. 

The  general  magnetic  field  of  the  sun  has  been  determined 
with  considerable  precision,  as  well  as  have  some  of  its  character- 
istics. Generally,  the  magnetic  field  has  opposite  signs  in  the 
two  hemispheres,  such  that  the  true  positive  magnetic  pole  is 
on  the  south  side  of  the  plane  of  the  ecliptic  on  the  southern 
hemisphere  of  the  sun  and  the  negative  magnetic  pole  is  on 
the  north  side,  exactly  as  is  the  case  with  the  distribution  of 
the  earth's  magnetism.  It  has  been  suggested  that  the  nega- 
tive ions  as  a  whole  are  more  distant  from  the  axis  of  rotation 
than  are  the  positive  ions,  and  that  the  rotation  of  the  sun 
produces  its  magnetism.  It  has  been  thought  that  the  internal 
magnetization  is  produced  by  ampere  electric  circuital  currents 
rotating  about  lines  which  are  somewhat  parallel  to  the  sun's 
axis  of  rotation,  as  the  inner  field  of  a  spherical  magnet  is  located. 
The  maximum  production  of  ions  is  near  the  bottom  of  the 
isothermal  layer,  and  the  maximum  of  disturbed  circulation  is 
also  near  that  level.  Hence,  magnetic  field  increases  from  a 
minimum  in  the  levels  above  the  photosphere  to  this  level. 
Whether  that  isothermal  value  is  a  true  maximum,  or  only  a 
step  toward  a  higher  value  in  the  lower  levels,  is  not  now 
known.  The  average  value  of  the  solar  magnetic  field  is  ap- 
proximately as  follows: 

Near  the  coronal  poles,      50  gausses 

In  small  sun-spots,          1000       " 

In  largest  sun-spots,       5000       " 

The  sun's  polar  magnetism  is  about  80  times  as  much  as 
the  earth's  polar  vertical  field,  which  is  0.66000  C.  G.  S.  units. 
From  such  data  the  entire  magnetic  system  of  the  sun  can  be 
readily  computed  by  the  well-known  formulas. 


OTHER  SOLAR  PHENOMENA  279 

It  has  been  found  that  the  Zeeman  Effect  indicates  that 
there  is  a  maximum  in  each  hemisphere  in  the  latitudes  ±  45°, 
and  that  a  sine  curve  expresses  the  distribution,  so  that  there 
is  zero-value  at  the  poles  and  at  the  equator.  By  referring  to 
Fig.  30  of  the  general  solar  circulation,  it  is  seen  that  the  maxi- 
mum vertical  circulation  has  been  placed  in  about  latitudes 
=*=  45°  in  each  hemisphere,  with  minimum  at  the  poles  and  at 
the  equator.  Hence,  it  may  be  inferred  that  there  is  a  sec- 
ondary magnetic  solar  field  due  to  vertical  circulation  in  latitude, 
as  if  the  ions  were  transported  upward  and  downward  in  these 
paths.  Very  similar  convectional  circuits  in  the  earth's  atmos- 
phere are  found  to  account  quite  fully  for  the  existing  diurnal 
magnetic  variations,  and  there  are  other  circuits  concerned 
with  the  general  disturbances  in  latitude. 

If  circulation  in  zonal  sheets  in  latitude  is  sufficient  to  cause 
a  persistent  maxima  in  certain  latitudes,  it  follows  that  a  similar 
segregated  circulation  in  longitude  is  sufficient  to  account  for 
the  maxima  of  the  magnetic  field  and  the  maxima  of  the  radia- 
tion itself,  arranged  in  longitude,  as  is  suggested  by  the  maxima 
of  Fig.  30.  There  are  persistent  and  unquestioned  impulses 
from  the  sun  operating  on  the  earth's  atmosphere  in  numerous 
manifestations,  which  indicate  that  there  is  a  general  periodic 
action  in  26.68  days,  upon  which  are  superposed  eight  distinct 
maxima  and  minima,  arranged  more  or  less  permanently  in 
longitude.  These  are  marked  most  clearly  in  the  variations  of 
the  terrestrial  magnetic  field,  and  less  emphatically  in  the 
meteorological  field.  It  is  stated  that  these  maxima  in  longi- 
tude are  not  found  in  the  spectrum  changes.  This  can  be 
readily  understood  from  the  fact  that  the  spectrum  observed 
belongs  generally  to  the  strata  lying  above  the  level  of  the  iso- 
thermal layer,  while  the  maximum  impulses  themselves  originate 
in  convectional  operations  which  are  rigorously  confined  below 
that  stratum.  The  sun-spots  penetrate  this  layer,  but  they 
have  never  been  found  to  synchronize  perfectly  in  their  fre- 
quency with  the  system  of  magnetic  pulses  that  are  observed. 
The  minor  variations  in  the  spectral  lines,  due  to  changes  in 
pressure,  density,  and  motion,  are  too  superficial  to  take 


280 

account  of  the  deep  sources  of  the  solar  radiant  energy  and  its 
allied  forces.  The  radiation  energy  of  the  sun  and  the  mag- 
netic field  of  the  earth  constitute  a  sensitive  mechanism  which 
registers  these  maximal  fluxes  of  energy,  and  it  would  be  re- 
quiring too  much  of  the  spectrum  to  attempt  to  see  in  it  a  full 
record  of  such  solar  actions.  The  interpretation  of  all  the 
symptoms  of  these  fields  will  doubtless  be  greatly  improved  by 
experience,  and  it  may  happen  that  some  telltale  signs  can  be 
discovered  which  will  serve  to  indicate  the  presence  of  maxima 
in  the  radiation.  At  present  the  negative  results  from  the 
spectroheliograph  should  not  be  interpreted  as  conclusive  evi- 
dence that  the  problem  of  maxima  has  been  exhausted  as  to  their 
synchronisms. 

The  Solar  Spectra 

It  is  not  our  purpose  to  study  the  characteristics  of  the 
solar  spectra  themselves  so  much  as  to  point  out  the  physical 
conditions  under  which  they  are  formed.  A  good  description  of 
them  may  be  found  in  Abbot's  "Sun,"  and  the  technical  articles 
are  collected  in  the  Astro  physical  Journal  from  the  Mt.  Wilson 
and  other  observatories.  The  following  summary  of  the  facts 
of  observation  is  derived  from  these  sources,  and  the  important 
matter  is  to  compare  them  with  the  data  of  the  preceding 
tables  of  solar  physical  conditions.  It  will  be  seen  that  a 'very 
large  addition  has  been  made  to  the  knowledge  of  such  condi- 
tions, and  that  while  the  general  harmony  between  the  ob- 
served and  computed  data  is  excellent,  there  will  be  needed 
many  modifications  in  the  inferences  that  have  been  made  from 
the  behavior  of  the  spectral  lines.  The  present  knowledge 
of  the  prevailing  temperatures,  pressures,  densities,  and  gas 
coefficients,  in  all  strata  for  gases  ranging  from  hydrogen  to 
mercurial  vapor,  and  the  heights  at  which  they  occur,  will 
greatly  improve  the  interpretation  of  the  reversals,  the  shifts, 
the  broadening  of  the  lines,  the  values  of  the  long  and  short 
lines  in  respect  of  height,  and  similar  problems.  It  will  be 
pointed  out  that  we  have  the  material  for  computing  the  coeffi- 
cients of  transmission  and  absorption  of  the  different  lines  of 


OTHER  SOLAR  PHENOMENA  281 

the  spectrum,  and  the  indices  of  refraction  in  all  layers  of  the 
atmospheres  of  the  earth  and  the  sun.  From  these  data,  and 
those  in  the  tables,  studies  can  be  made  in  atomic  and  molecular 
physics,  under  the  conditions  of  solar  temperature  and  gravi- 
tation, and  thus  escape  from  many  laboratory  limitations. 


The  General  Solar  Spectrum 

The  solar  spectrum  consists  of  a  continuous  background, 
crossed  by  dark  lines  which  are  commonly  identified  with  those 
of  the  terrestrial  elements.  The  spectra  of  all  gases  become 
continuous  at  high  temperatures  and  high  pressures.  By  Table 
6  the  temperatures  in  the  isothermal  layer  range  from  7650° 
to  7700°,  and  in  the  adiabatic.  layer  they  increase  rapidly  to 
enormous  values;  by  Table  10  the  pressures  in  terrestrial  at- 
mospheres are  about  6.08  at  the  photosphere,  about  20  atmos- 
pheres at  the  bottom  of  the  isothermal  layer,  and  they  soon 
become  enormous  in  the  adiabatic  strata;  by  Table  12  the 
density  for  hydrogen  is  ^  that  of  the  earth's  normal  atmos- 
phere at  sea  level,  and  mercury  vapor  has  Vio  that  density, 
while  at  the  bottom  of  the  isothermal  layer  its  density  is  %  that 
of  air,  so  that  terrestrial  sea-level  densities  occur  only  in  the 
adiabatic  layer,  beyond  the  vision  of  the  spectroscope;  the  gas 
efficiencies  are  indicated  in  Table  13.  These  are  the  conditions 
for  the  continuous  spectrum.  In  the  reversing  layer  the  tem- 
peratures of  the  light  gases,  2  to  60  atomic  weight,  have  been 
lowered  by  2000  to  3000  degrees,  while  the  heavy  gases  and 
vapors  do  not  extend  to  the  altitude  of  400  to  500  kilometers. 
These  dark  lines  are  formed  by  interposing  a  cooler  absorbing 
layer  of  the  same  gas  between  the  continuous  spectrum  and  the 
spectroscope.  Most  of  the  dark  lines  are  solar,  as  iron  (55), 
nickel  (58),  calcium  (40),  titanium  (48),  cobalt  (59),  chromium 
(52),  magnesium  (24),  carbon  (12),  vanadium  (51),  sodium  (23), 
magnesium  (24),  hydrogen  (2),  these  being  the  lines  just  before 
reaching  the  axis  of  the  curves  of  Fig.  18.  The  heavy  vapors 
and  gases  are  seen  with  difficulty,  only  by  intrusion  above  their 
normal  levels.  In  the  earth's  atmosphere  there  are  dark  lines 


282  A   TREATISE   ON  THE   SUN'S   RADIATION 

due  to  absorption  at  very  low  temperatures  by  oxygen,  car- 
bonic acid,  and  aqueous  vapor. 

Tests  are  used  to  determine  whether  a  line  is  solar  or  ter- 
restrial: (1)  at  high  and  low  sun,  the  terrestrial  lines  are  stronger 
at  low  sun;  (2)  the  east  and  west  limbs  of  the  sun  directed 
simultaneously  on  the  slit  give  the  Doppler  Effect  of  shift  for 
solar  lines,  while  the  terrestrial  lines  remain  unchanged. 

The  principal  lines  of  the  spectrum  are  marked,  in  Ang- 
strom units,  namely,  tenth-meter  =  0.000  000  000  1  meter. 

K  3933.68  calcium,       solar. 

H  3968.49  calcium, 
Hy  =  G  4340.47  hydrogen        i( 
Hp  =  F  4861.35        "  " 

b  5183.62  magnesium     " 

E  5269.55  iron  " 

Z>2  5889.98  sodium  " 

Ha  =  C  6562.84  hydrogen      • " 

B   6869.97  oxygen      terrestrial 
a  7184.57  aqueous  vapor  " 

A    7593.83  oxygen 

The  intensities  of  the  lines  are  classified:  ^ 

0000  =  the  most  difficult  to  see. 

1  =  just  clearly  visible  on  Rowland's  spectrum  map. 

1000  =  the  strong  calcium  lines,  H.  K. 

In  the  solar  spectrum  the  intensities  decrease  with  the 
increase  in  the  atomic  weight,  till  radium  (224)  and  uranium 
(236)  may  exist  in  the  sun  without  being  seen  in  the  lines; 
there  are  only  a  few  non-metallic  elements;  oxygen  and  helium 
exist;  a  very  little  absorbing  gas  produces  a  dark  line;  lines 
are  dark  only  by  contrast,  so  that  the  flash  spectrum  of  the 
reversing  layer  consists  of  bright  lines  in  the  places  of  dark 
lines,  as  it  is  seen  in  eclipses  at  the  moment  the  photosphere 
disappears;  short  waves  are  more  absorbed  than  long  waves, 
but  the  lines  from  1.50  n  to  2.50  p  pass  through  the  intervening 
layers  of  the  solar  and  terrestrial  atmospheres  with  only  selec- 
tive band  absorptions,  while  the  short  waves  from  0.00 /z  to 


OTHER  SOLAR  PHENOMENA  283 

0.35  /x  generally  have  disappeared;   the  intermediate  lines  0.35  ju 
to  1.50  M  are  depleted  in  an  irregular  manner. 

Whatever  changes  the  temperature,  pressure,  and  density, 
relatively  to  certain  normal  values  on  the  different  levels, 
also  causes  variations  in  the  position  and  shape  of  the  lines. 
Thus,  convection  by  temporary  transportation  changes  these 
normal  relations  in  vertical  directions,  while  circulation  changes 
them  in  horizontal  directions,  so  that  on  the  sun  these  varia- 
tions are  numerous  and  very  complex. 

Increase  of  pressure  shifts  the  wave  lengths  toward  one 
end  or  other  of  the  spectrum,  and  it  broadens  certain  lines. 
On  the  whole,  the  shifts  increase  with  the  wave  length,  but 
there  are  many  arbitrary  conditions.  On  the  other  hand,  change 
in  velocity  shifts  all  lines  proportional  to  their  wave  lengths. 
Due  to  the  rotation  of  the  sun,  all  solar  lines  have  a  little  greater 
wave  length  than  the  corresponding  terrestrial  lines  by  a  few 
thousandths  of  an  Angstrom.  The  general  pressure  is  from  4 
to  6  atmospheres,  as  determined  by  the  spectrum.  There  is 
unsymmetric  broadening  of  some  lines,  and  greater  shift  on  the 
side  of  the  long  waves  than  on  the  side  of  the  short  waves. 
There  is  a  common  vertical  circulation  upward  in  general  of 
0.1  to  0.3  kilometers  per  second;  there  are  large  vertical  move- 
ments in  the  granulations  and  pores,  in  the  faculae  arid  sun- 
spots,  and  they  become  at  times  excessive  in  the  solar  promi- 
nences. These  can  all  be  discussed  by  the  formulas  of  this 
Treatise. 

Lines  are  also  classified  by  their  temperatures: 
Enhanced  lines  =  high  temperatures  =  spark  conditions, 
Average  lines     =  low  temperatures  =  arc  conditions. 

The  enhanced  lines  generally  indicate  vertical  convection, 
bringing  high-temperature  conditions  upward  as  in  the  umbra 
of  spots,  faculae,  and  granulations;  low-temperature  conditions 
occur  over  the  center  of  sun-spots  in  the  reversing  layer,  in 
the  penumbra  and  in  the  pores  between  the  filaments  and  the 
granules  wherever  there  is  downward  circulation;  irregular 
mixtures  occur  in  all  levels  due  to  horizontal  movements. 

The  chromosphere,  especially  in  the  reversing  layer,  has  a 


284  A   TREATISE   ON  THE   SUN'S   RADIATION 

spectrum  opposite  to  that  of  the  photosphere;  there  are  double 
reversals  in  some  lines,  as  3933.667,  wherein  KI  is  probably 
stationary,  while  K%,  K%  has  a  vertical  velocity  upward  of  1.97 
kilometers,  and  K3  a  downward  velocity  of  1.14  kilometers  per 
second.  These  represent  layers  of  different  levels,  KI  lowest, 
K2  higher  and  rising,  K9  highest  and  falling.  In  the  sun- 
spots  the  vortex  circulations  bring  enhanced  lines  from  below 
upward  into  the  vortex  of  the  penumbra;  the  high  level  gases  of 
the  chromosphere  descend  with  their  own  lower  temperatures, 
as  in  the  penumbra  and  the  reversing  layer;  there  are  lines 
from  layers  still  somewhat  normal,  so  that  the  sun-spot  spectra 
are  very  complex  and  differ  from  that  of  the  surrounding  photo- 
sphere as  indicated.  Seen  in  superposition,  there  are  many 
phenomena,  such  as  increase  in  the  shadings  and  wings;  en- 
hanced lines  apparently  weaker  in  spots  through  lowering  of 
temperatures  from  higher  strata;  Ha  weaker  in  spots;  maxi- 
mum ordinate  of  radiation  shifts  from  the  penumbra  to  umbra 
with  the  change  in  the  temperature;  the  short  waves  are  rela- 
tively much  weaker  in  the  umbra  than  in  the  penumbra  and 
the  ratio  approaches  1.000  in  the  long  waves;  temperature  over 
sun-spots  less  than  that  of  the  surrounding  photosphere;  more 
scattering  and  absorption  above  spots  than  above  the  photo- 
sphere; many  Fraunhofer  lines  are  strengthened  and  many 
are  weakened  in  the  sun-spots  as  compared  with  the  photo- 
sphere; sun-spot  vapors  are  too  cool  to  produce  strong  absorp- 
tion of  the  enhanced  lines;  great  abundance  of  flu  tings  in  the 
sun-spots;  high  temperatures  produce  complete  dissociation  in 
the  lower  part  of  the  sun-spot  vortex,  but  tendency  to  associa- 
tion in  the  central  part  of  the  isothermal  layer  and  an  abun- 
dance of  associated  compounds  above  the  photosphere  itself; 
reduction  of  the  continuous  background  in  spots  is  greatest  for 
the  short  waves. 

As  between  the  spectra  at  the  center  of  the  disk  and  the 
limb  of  the  sun,  the  lines  are  generally  displaced  toward  the 
red  on  account  of  an  increase  in  total  pressure  under  hemi- 
spherical curvature.  Hydrogen  (2),  sodium  (23),  calcium  (40), 
magnesium  (24)  show  no  displacement;  titanium  (48),  vana- 


OTHER  SOLAR  PHENOMENA  285 

dium  (51),  scandium  (44)  show  moderate  displacement;  iron 
(55),  nickel  (58)  show  considerable  up  to  0.007  Angstroms. 
That  is,  high  atomic  weights  are  but  little  displaced;  enhanced 
lines  show  maximum  displacement.  The  violet  edges  of  the 
lines  do  not  shift.  The  limb  spectra  are  weaker  than  at  the 
center,  and  the  violet  lines  need  much  more  exposure  at  the 
limb;  the  Fraunhofer  lines  are  much  changed,  especially  in 
the  violet;  the  strong  lines  lose  their  shading  or  wings;  the 
enhanced,  high  temperature  lines  are  weakened  at  the  limb; 
the  lines  which  are  strong  in  spots  are  strong  at  the  limb;  the 
hydrogen  Ha  is  widened  at  the  limb. 

Atmospheric  Refraction  and  Scattering 

The  general  relations  between  the  refraction  or  change  of 
direction  of  a  ray  of  light  passing  through  an  atmosphere  and 
the  loss  of  the  energy  by  non-selective  scattering  upon  the 
molecules  are  expressed  by  Rayleigh's  Formula: 

,  32T»(tf-l)«fc£      327r3(M-l)2  .  J*_ 

3  X4  n0  .  Bo  3  X4  go  Wo  Po 

in  the  following  notation, 

k  =  coefficient  of  scattering,  /*  =  index  of  refraction, 
X  =  the  wave  length  in  centimeters,  lo  =  the  height  of  the 
homogeneous  atmosphere  at  the  stratum  whose  pressure  is  B0, 
B  =  the  barometric  pressure  at  the  level  under  discussion, 
n  =  the  number  of  molecules  per  cu.  cm.,  System  =  (C.  G.  S.) 

/IOP\        «r     i  loB.  pm  BQ        B  pm  B 

(185)     We  have,  -=-  =  --  •  —  =  --  = 

JJQ  PO  -t>0  PO 


Hence,  by  transformations, 
(186)  (M_1)2=^ 


P  1 

--  —  ,  Table  3,  Treatise. 

go       Po 


On  the  sea  level,  from  which  the  homogeneous  height  may 
be  computed,  or  on  the  photosphere  of  the  sun,  the  pressure 
P  =  PQ.  The  formula  was  devised  to  compute  k  at  any  height 
above  the  plane  of  reference,  but  since  we  have  computed 


286 


A   TREATISE   ON  THE   SUN'S  RADIATION 


w,  p,  P,  at  numerous  points  in  each  atmosphere,  we  can  at  once 
take  them  at  the  point  in  any  stratum  of  any  gas  for  which 
they  have  been  prepared.  The  conditions  are,  however,  so 
complex  for  the  different  wave  lengths  that  it  is  possible  merely 
to  give  some  approximate  results  in  the  atmospheres  of  the 
earth  and  the  sun  as  examples  of  a  method  that  can  be  utilized 
in  discussing  these  important  subjects.  It  will  be  necessary  to 
obtain  the  values  of  k  from  the  observations  with  the  pyrheliom- 
eter  and  the  bolometer,  and  it  will  be  convenient  to  summarize 
the  formulas  of  the  theory  of  refraction. 

THE  FORMULAS  OF  REFRACTION,  FIG.  28,  IV 


Terms 

Polar 
Coordinates 

Angle 
Incidence 

Angle 
Refraction 

Refraction 

Index  of 
Refraction 

Upper  point  

(R  +  z),  (0+<#) 

ili  iz  •  •  • 

/i,  /i.  • 

S,,  52  .  .  . 

Mi»  /*2  •  «  • 

Lower  point 

R   0 

z 

<p  =  the  angle  which  the  tangent  to  the  light-curve  in  the 
plane  xy  makes  with  the  axis—  x.  f  =  the  zenith  distance. 

a  =  the  angle  of  the  ray  with  an  horizon  =  90—  /. 

R  =  the  radius  to  the  lower  level,  R  +  z  to  the  higher 
level. 

6  =  the  angle  from  x  to  R,  6  +  d  0  =  the  angle  from 
*  to  R  +  z. 

(187)   x  =  r  cos  0,  generally,    d  x  =  d  s  cos  <?.     j-  = 

Cl  S 

de   .          ndr 

—  r  sin  0  3  —  h  cos  6  3—. 

as  as 

d  y 
y  =  r  sin  6,  generally,    d  y  =  d  s  sin  <p.     j-  = 

(/  -S 

dd  ,  dr 

+  r  cos  0-3  —  h  r  sin  0  3—. 

d  s  d  s 


™  *' 
cidence. 


—  0  =  90°  —  a  =  the  angle  of  refraction. 
-0  +  S  =  90-a  +  S  =  the  angle  of  in- 


OTHER   SOLAR    PHENOMENA  287 


(189)  From  the  laws  of  optics,—  =  - 


sin/ 

(190)  From  the  triangle  of  (R,  R  +  z),     R  *  *•  j  j~7  •     Hence, 

(191)  sin/  =  -  sin  *  =  -TTi  —sin  {•,  and  generally, 

Mo  •*»-    I 

(192)  ft  (R  +  z)  sin  *  =  MO  R  sin  f  =  C  =  constant. 

—  M       sin  i  —  sin/       tan  J  (*'—/) 


(193)  From  (189) 


770  +  M       sin  i  +  sin/       tan  J  (*'  +  /) 

i£i 

5  tan* 


(194)  The  differential  equation  of  refraction,  d  d  =  —  tan  i. 

(195)  Hence  M  cos  i  .  d  8  =  d  /*  .  sin  *". 

More  analytically,  we  have  to  quote  the  differential  equation 
(Kummer,  Sitz.  Berlin  Akad,  12  July,  1860), 


Substituting  the  auxiliaries,  there  are  two  equations, 

(197)  I.     M  (r  cos  6  .  sin  <p  —  r  sin  0  .  cos  <p)  =  vr  sin  (^  —  0)  = 

/*/•  sin/=  C 

(198)  II.    /*  y  cos  0  (r  cos  0  j^  +  sin  0  j^)  -  p  r  sin  0  (—  r  si 


sn 


.  ,  - 

01-  +  cos0T-)=Mr2^-  =  C 
(/5  ds*  ds 

(199)  M  r2  d  d  =  C  d  5  =  C  (d  r*  +  r*  d  &*)*,    d  B  =  r(^r,_ 
rz  Mo  R  sin  T  d  z 


(R  +  2)  U2  (*  +  *)2  -  M2o  ^2  si 
(201)  There  are  three  cases,     I.  M  (R  +  z)  >  fr  R  sin  £, 

atmospheric  refraction 
II.  /i  (12  +  z)  =  /IQ  ^  sin  r, 

circular  refraction 

III.  A«(^  +  2)  <  ^o^sinr, 

imaginary  refraction. 


288  A   TREATISE   ON  THE   SUN'S  RADIATION 

The  index  of  refraction  is  connected  with  the  density  of  the 
stratum  by  the  general  equation, 

/onoN    M2  —  1  =  4  K  p  ..dn       2  K  .  d  p  , 

(202)  .   n        and  —  =  ,  .  ,      ,  for  K  =  a  constant. 

/cp' 


Several  hypotheses   (Chauvenet)   have  been  employed   to 
determine  p,  the  density  in  the  strata  above  the  sea-level  value,  p0. 

r>  n  7?  T* 

(203)  —  =      p  ^  ,  by  the  Boyle-Gay  Lussac  Law. 

FQ         po  AO  1  0 

I.  It  is  throughout  these  discussions  assumed  that  R  = 
RQ  =  constant,  and  consequently  it  is  erroneously  supposed 
that  the  atmosphere  is  stratified  along  adiabatic  gradients. 
This  being  the  case,  the  discussions  need  complete  revision 
throughout  the  formulas.  This  is  an  exceedingly  complicated 
problem,  and  it  may  prove  that  the  empirical  formula  of  Bessel 
cannot  be  improved  for  practical  purposes.  The  first  supposi- 
tion regarding  the  temperature  is  that  T  =  T0,  and  is  isothermal. 
Hence, 

(204)  L  =  P-  =  e~l 

FQ         po 

This  does  not  lead  to  satisfactory  results. 

II.    Assume  the  following  relations,  which  are  easily  found: 


~          R' 
This  finally  leads  to  the  equation, 

(206)  F  '  -  =  F  =  x  -  2T- 

ro       p         lo  2  to 

It  can  be  easily  shown  that  the  temperatures  do  not  con- 
form to  this  formula  in  the  higher  strata. 

III.  Assume,  with  Bessel,  the  following  fundamental 
formula, 

(207)  =  e-f  =  l--  +  ii2-...=          Hence, 


PO 


OTHER    SOLAR   PHENOMENA 


289 


Take  h  =  7991.04,  and  h  =  227775.7  meters.     In  common 
logarithms, 

(209)    log  P  =  log  po  -  M  (|  -  -J-). 


TABLE  92 

THE  DENSITY  p  AS  COMPUTED  BY  THE  NON-ADIABATIC,  THE  ADIABATIC, 
AND  THE  BESSEL  FORMULAS 

Balloon  Ascension,  Uccle,  June  9,  1911 
(210)  Non-adiabatic.    Log  pi  =  log  po  +  ~~T  (log  Ti  -  log  T0) 

i\    ~—    J. 


(211)  Adiabatic. 

(212)  Bessel. 


Log  pi  =  log  Po 

Log  pi  =  log  Po  -  M  \       - 


Height  in  Meters. 

Non-adiabatic 
(Bigelow) 

Adiabatic 

Bessel 

2  =  50000  .  .  . 

0.0009 

0  0003 

0.009 

45000  

0.0054 

0.0022 

0.053 

40000 

0  0160 

0  0049 

0  097 

35000.  .  . 

0  0293 

0.0088 

0.177 

30000  

0.0511 

0.0188 

0.0324 

25000 

0  0885 

0  0407 

0  0593 

20000...      ... 

0.  1543 

0.0901 

0.1084 

15000 

0  2708 

0  2012 

0  1983 

10000.  .  . 

0  4753 

0  .  4179 

0.3627 

5000.  

0.7830 

0.7229 

0.6633 

000.  . 

1  2132 

1  2132 

1  2132 

Taking  the  same  value  of  po  =  1.2132  at  the  sea  level,  it  is 
seen  that  the  adiabatic  is  lower  in  amount  than  the  non-adia- 
batic  value  up  to  50000  meters;  the  Bessel  value  is  lower  than 
the  non-adiabatic  throughout;  it  is  less  than  the  adiabatic 
from  the  sea  level  to  15000  meters,  and  then  remains  interme- 
diate between  the  adiabatic  and  the  non-adiabatic  values  up 
to  50000  meters.  It  will  be  exceedingly  difficult  to  develop 
any  formula  that  can  supply  the  place  of  the  non-adiabatic 
temperature  gradient,  while  depending  solely  upon  the  surface 
conditions.  In  the  case  of  the  Bessel  formula  for  refraction, 
upon  which  the  working  tables  are  constructed,  it  is  probable 


290  A  TREATISE   ON  THE   SUN'S  RADIATION 

that  the  exponents  A  and  X  make  some  compensation  for  the 
actual  inaccuracy  in  the  density  p. 

(213)  B  =o  0*  TA  tan  f  =  Bessel's  Formula. 

r/?e  Atmospheric  Transmission  for  Different  Wave  Lengths  pK 
Collected  According  to  the  Value  of  pw  by  the  Pyrheliometer 

Volumes  II  and  III  of  the  Annals  of  the  Astrophysical  Obser- 
vatory of  the  Smithsonian  Institution  contain  the  coefficients 
of  transmission  as  measured  by  the  pyrheliometer  pw,  together 
with  the  coefficients  for  several  wave  lengths  as  determined 
by  the  bolometer  p^.  These  have  been  collected  in  groups 
with  pw  Sit  convenient  intervals,  0.900  —  0.890,  0.890  —  0.880, 
0.880  —  0.870,  etc.,  and  the  mean  values  were  obtained  for  the 
several  years.  These  were  united  in  general  means,  and  they 
were  plotted  on  large  diagrams,  with  p^  for  ordinates  and  X 
the  wave  length  for  abscissas.  Each  group  under  the  pyrheli- 
ometer pw  results  in  a  curve  of  transmission  coefficients,  and 
the  series  of  pw  make  a  family  of  curves.  These  were  adjusted 
to  probable  values,  and  from  them  were  scaled  the  final  values 
which  appear  in  Tables  93,  94,  95,  for  Washington,  Mt.  Wilson, 
and  Mt.  Whitney,  respectively. 

Resuming  the  black  spectrum  for  6950°,  corresponding  with 
3.98  gr.  cal./cm.2  min.,  the  spectrum  values  for  the  different 
wave  lengths  were  multiplied  by  the  coefficients  of  transmis- 
sion under  the  several  values  of  pw,  resulting  in  the  groups  of 
pi  and  /A.  The  latter  represent  the  transmitted  values  of  the 
radiation,  and  as  they  are  homogeneous  throughout  they  give 
relative  values.  Take  the  sums  for  the  selected  wave  lengths 
along  the  spectrum;  the  factor  16.7  suffices  to  reduce  these 
sums  to  calories,  and  it  is  seen  that  for  each  value  of  pw  there 
are  corresponding  calories, 
such  as,  2.79,  2.71,  .  .  .  2.20,  2.13  for  Washington, 

"        2.95,  2.92,  .  .  .  2.72,  2.63  for  Mt.  Wilson, 

"        3.04,  2.99,  .  .  .  2.78,  2.72  for  Mt.  Whitney. 
The  pyrheliometric   transmission   at   different  values   has   its 
counterpart  in  the  bolometric  transmission  at  different  wave 


OTHER    SOLAR   PHENOMENA 


291 


PH    H 


w  S     a 

g<    2 
bo 

p| 

<   Q     J> 
fc    W 


go 

Q 


K^  I 


I  S 

|  § 

•S  ii 

J3 


»-H»c<ioecoOrHOONecNNOi'-it-«cooM«ooi 


f^C^^^OCDtDt^ 


ONoootocowco 


Oi  ^H  t— 

CO  t-N 


<THC»OiO>5OOSCOC-COOT}"iOOOOOieOt-O3O>t--^NOO500 
IO  O  T-H  O  00  i-(  IO  O>  U5  d  O5  t-  «D  't  CO  CO  C<1  N  <-H  rH  i-(  rH  iH  O  O 


t-  1-  1-  oo  oo  oo  o>  oj  o>  cj  cs  Oi  os 


T-li-IC^COI 

oj  o>  cj  cs 


Sliiigiiiig 


oo  oo  oo  os  os  os  os 


T}<Tl<-^<TtTj<^}iTl<r}irJ<T}iTfT}<TtiT)<-^ 

os  os  os  os  os  os  os  o>  os  os  os  os  os  os  os 


TfeoeoocccDOC^M-^icioioiommiflioiflkOioioiomious 

<  Ui  «O  t>  OO  00  00  OS  OS  OS  OS  OS  OS  OS  OS  OS  OS  OS  OS  OS  OS  OS  OS  OS  OS  OS  OS 


OSU5C<lNiOI-CC<lOOO"5«OOST-tOSrC^Hi-(iOO0^1<Ot>Tj<C<lTHOOS 


NOOco-«i|oo3co«ot-co  t-'oo  t-coinc 

t^t^CDt^C^COOOC^TfiOCDCDCDCD^O^DC 


JOlOkOrHlOOOOOSINI 


5  §  s 


OOOOOOOOOOOi 

I 

§ 


iMCvJNdNNN 


faU 


292 


A   TREATISE   ON   THE   SUN  S   RADIATION 


H       c 

S    o 

*  S 

u    ^ 

51 

a  ^ 

H      ^ 

K   S 

I 

o 

w 

B 


I    »H  o  Tf  oo  as  lo-cv)  ic  co  oo  oo  I-H  co  «3  t>  oo  as  o  N  co  ••*  10  w  t-  oo  os  < 
•^5,     '  ^t  10  <o  t>  c- oo  oo  oo  Oi  05  as  a>  o>  os  as  o  as  os  os  as  as  Oi  os  a>  as  as  < 


T)«O^-^'OSDO5O5Tl<»HC<HOa>OOO3O 


.   OSTt^0^05t2coSt-2S(NO>t--Si3^«NwS! 
iH  CO*  Tj<  Tj<  Tj<  Tj«  Tji  CO*  N  N  r-i  iH 


m«£>io<Doooot-oorfTtit-oo5coT-i^-iinoo-^ot-Tiieo»-ioa> 


'         COC^t^ 


eCeOO- 
'   ^*COt— 


JINN(NC<l 


«cooooo>t--^»NTi'c<it-t-O5NO'^Noa>ooo-^<o 


Ui^l'1^<T-l«DSOCDT}<C»OOOC<IOiOC<lNinoO>O»-HOOlOCOi-lOOS 


00  Oi  Oi  O>  OJ  OJ  O5  O5  O5  OS  O5  05  OS  O3  OS  Oi  OS  OJ  OS  O>  OS  OJ 


cD»-iO5oom^Ha5Oix>oiooi-ic<i'-iioc<iwoc5iorH 


Tf  CO  N  CJ  rH  »H  i-l 


IO«OOOOOCO«OOOOtD"3O5CO'*^}<'^^-^t'^ 

1  o^-^Nt-Or-iiocot-t-oooooooooooooo 
i  T-o5sjs>5iOiO5a5aiOi 


^ 


OOOOOOOOOOOi 

I 


*  N  CS1  N  W  N  M  d 


OTHER    SOLAR    PHENOMENA 


293 


lengths.  This  group  of  results  conforms  in  their  average  values 
to  those  which  have  been  heretofore  employed,  Table  79,  but 
they  express  in  detail  the  effects  of  the  atmospheric  conditions 
upon  the  spectrum  processes  of  scattering. 


TABLE  95 

THE  COEFFICIENTS  OF  TRANSMISSION  AND  THE  ORDINATES  IN  THE  SPECTRUM 
FOR  DIFFERENT  VALUES  OF  THE  PYRHELIOMETER  pw 

Mt.  Whitney,  4420  meters 


Pyrheliometer 

t-w 

0.938 

0.918 

0.898 

0.878 

0.858 

0.838 

r  =  6950° 

A 

P*     'A 

*A     'A 

P*     A 

*A     J\ 

*A    'A 

X       A 

OftO      0   9^iL 

7  3^ 

.  uu  —  u  .  ZD/J-  .  .    .  . 
0.30   .  . 

I  .  oO 

4.85 

.510 

2.47 

.508 

2.46 

.506 

2.45 

.504 

2.44 

.502 

2.43 

.500 

2.43 

0.35   .... 

6.05 

.670 

4.05 

.631 

3.82 

.595 

3.60 

.574 

3.47 

.556 

3.36 

.545 

3.30 

0  .  40   .  . 

6.54 

.810 

5.30 

.772 

5.05 

.743 

4.86 

.700 

4.58 

.665 

4.35 

.642 

4.20 

0.45 

6.50 

.877 

5.70 

.854 

5.55 

.826 

5.37 

.792 

5.15 

.772 

5.02 

.750 

4.88 

0.50   .... 

6.13 

.913 

5.60 

.898 

5.50 

.878 

5.38 

.852 

5.22 

.827 

5.07 

.804 

4.93 

0.55   .. 

5.59 

.930 

5.20 

.919 

5.14 

.904 

5.05 

.883 

4.94 

.861 

4.81 

.837 

4.68 

0.60    ..       . 

5.00 

.942 

4.71 

.932 

4.66 

.921 

4.61 

.907 

4.54 

.884 

4.42 

.859 

4.30 

0.70    ..       . 

3.88 

.961 

3.74 

.950 

3.70 

.941 

3.66 

.929 

3.61 

.911 

3.54 

.890 

3.46 

0  .  80 

2.96 

.971 

2.87 

.962 

2.85 

.952 

2.82 

.941 

2.79 

.927 

2.74 

.909 

2.69 

0.90   . 

2.25 

.976 

2.20 

.967 

2.18 

.958 

2.16 

.947 

2.13 

.934 

2.10 

.921 

2.07 

1.00   . 

1.72 

.978 

1.68 

.970 

1.67 

.962 

1.65 

.952 

1.64 

.939 

1.62 

.926 

1.59 

.10   . 

1.33 

.979 

1.30 

.972 

1.29 

.963 

1.28 

.953 

1.27 

.943 

1.25 

.930 

1.24 

.20   . 

1.04 

.980 

1.03 

.972 

1.02 

.964 

1.01 

.955 

1.00 

.946 

.99 

.931 

.98 

.30   . 

0.82 

.980 

.80 

.972 

.80 

.964 

.79 

.956 

.78 

.947 

.78 

.932 

.76 

.40   . 

0.66 

.980 

.65 

.973 

.64 

.965 

.64 

.956 

.63 

.947 

.63 

.932 

.62 

.50   . 

0.53 

.980 

.52 

.973 

.52 

.965 

.51 

.956 

.51 

.947 

.50 

.933 

.49 

.60   . 

0.43 

.980 

.42 

.973 

.42 

.966 

.42 

.957 

.41 

.947 

.41 

.934 

.40 

.70   . 

0.36 

.980 

.35 

.973 

.35 

.966 

.35 

.957 

.34 

.948 

.34 

.935 

.34 

.80   . 

0.29 

.980 

.28 

.973 

.28 

.966 

.28 

.957 

.28 

.948 

.27 

.936 

.27 

.90   . 

0.25 

.980 

.25 

.973 

.24 

.966 

.24 

.957 

.24 

.948 

.24 

.937 

.23 

2.00   . 

0.21 

.980 

.21 

.973 

.20 

.966 

.20 

.958 

.20 

.948 

.20 

.938 

.20 

2.10   . 

0.18 

.980 

.18 

.973 

.18 

.966 

.17 

.958 

.17 

.948 

.17 

.938 

.17 

2.20   . 

0.15 

.980 

.15 

.973 

.15 

.966 

.14 

.958 

.14 

.949 

.14 

.939 

.14 

2.30    . 

0.13 

.980 

.13 

.973 

.13 

.966 

.13 

.958 

.12 

.950 

.12 

.940 

.12 

2  .  40 

0.11 

.980 

.11 

.973 

.11 

.967 

.11 

.958 

.11 

.950 

.10 

.941 

.10 

2  .  50    .  .       . 

0.10 

.981 

.10 

.973 

.10 

.967 

.10 

.958 

.10 

.951 

.10 

.942 

.09 

2  .  60   .  .       . 

0.09 

.981 

.09 

.973 

.09 

.967 

.09 

.959 

.09 

.951 

.09 

.942 

.08 

65.50 

50.09 

49.10 

48.07 

46.90 

45.79 

44.76 

Factor  16.7 

765 

750 

734 

716 

.699 

.683 

3.98 

3.04 

2.99 

2.92 

2.85 

2.78 

2.72 

The  Terrestrial  Values  of  the  Index  of  Refraction  (n  —  1) 

We  shall  utilize  these  coefficients  of  transmission  p^  first  to 
compute  the  coefficients  of  absorption, 


(214) 


294  A   TREATISE   ON  THE   SUN'S   RADIATION 

and  then  proceed  by  the  Rayleigh  Formula  (184)  to  compute 
the  values  of  (/*  —  1)  for  the  index  of  refraction, 


From  Tables  93,  94,  95,  interpolate  for  four  selected  wave 
lengths  0.323  /i,  0.481  /*,  0.670  /i,  1.225  /i,  as  examples,  the  values 
of  p\  corresponding  to  pw.  Compute  the  equivalent  coeffi- 
cients of  scattering.  In  the  Rayleigh  Formula  we  have  for 
0.323  /i,  X4  =  (0.0000323  cm.)4;  g0  is  the  acceleration  of  gravita- 
tion 980.6;  n  is  the  number  of  molecules  per  cubic  centimeter, 
Tables  15,  21,  27,  note  page  55,  Bui.  No.  4,  0.  M.  A.;  p  is  the 
density  from  Tables  9,  17,  23;  P  is  the  pressure  from  Tables 
9,  17,  23.  The  mean  values  for  these  three  balloon  ascensions 
were  used  in  this  formula.  The  resulting  values  of  (/*  —  1) 
for  Washington,  Mt.  Wilson,  and  Mt.  Whitney  are  given  for 
four  wave  lengths,  in  order  to  illustrate  the  variability  in  the 
value  of  (/*  —  l).  It  increases  from  large  values  of  pw,  the 
pyrheliometric  coefficient  of  transmission,  to  low  values  of  pw\ 
it  increases  always  with  the  increase  in  the  wave  length;  it 
diminishes  generally  with  the  altitude  of  the  station. 

The  Solar  Values  of  the  Index  of  Refraction 

The  data  are  taken  for  the  computations  from  Abbot's 
Table  55,  Vol.  III.,  which  gives  the  relative  brightness  for  dif- 
ferent lines,  from  the  center  to  the  limb  at  several  radial  dis- 
tances. These  distances  are  a  =  r  sin  f  ,  from  which  the  zenith 
distance  £  and  the  sec  £  are  computed.  In  the  sun's  atmos- 
phere we  can  take  the  path  length  of  the  ray,  as  in  Section  IV, 
Fig.  28,  proportional  to  the  sec  £,  so  that  by  the  Bouguer  Formula, 


sec      —  sec 


In  order  to  compute  70,  /i,  /2,  .  •  •  the  relative  intensity, 
we  proceed  as  follows:  Take  the  relative  brightness  from  Table 
55;  interpolate  in  the  black  spectrum  of  7655°  for  the  given 
wave  length,  and  divide  by  20.9,  in  this  distribution  of  the 


OTHER    SOLAR    PHENOMENA  295 

spectrum  line  distances,  to  reduce  to  gr.  cal./cm.2  min.,  and 
this  is  the  assumed  value  of  70  at  the  center  of  the  disk;  mul- 
tiply 70  by  the  observed  brightness  at  the  other  radial  distances, 
so  that  we  have  the  successive  values,  70,  7i,  72,  .  .  .  ;  these 
were  plotted  in  pairs  on  a  diagram  whose  abscissas  are  sec  f 
and  whose  ordinates  are  log  7,  as  in  the  usual  pyrheliometric 
reductions;  the  resulting  values  of  pK  appear  in  the  Section  I, 
Table  97.  From  these  values  of  p^  compute  k^  in  Section  II. 
From  k^  compute  (/*  —  1)  in  Section  III,  for  four  wave-lengths 
of  hydrogen,  carbon,  calcium,  mercury.  The  gravity  accelera- 
tion G  =  27484.3  cm./sec.;  n  =  the  number  of  molecules  per 
cu.  cm.,  Table  32;  p  is  the  density,  Table  12;  P  is  the  pressure, 
Table  9. 

It  is  seen  that  (/*  —  1)  decreases  from  the  center  to  the  limb 
of  the  sun;  it  increases  with  the  wave  length;  and  it  increases 
with  the  molecular  weight.  The  terrestrial  values  of  (/z  —  1) 
in  the  lower  strata  of  the  atmosphere  are  about  ten  times  as 
large  as  they  are  for  the  solar  gases  in  the  neighborhood  of 
the  photosphere.  The  values  of  (M  —  1)  decrease  gradually 
to  vanishing  values  in  the  outermost  strata  of  both  atmospheres. 

General  Remarks 

There  can  be  no  greater  hardship  than  to  present  the  results 
of  an  extensive  logarithmic  computation  by  means  of  a  few 
selected  numerical  values  at  the  end.  The  entire  working  of 
the  formulas  in  their  details  is  lost  upon  the  reader,  and  the 
relations  are  in  themselves  so  complex  that  it  is  quite  im- 
possible to  follow  them  intelligently  apart  from  the  actual 
processes  that  are  involved.  In  this  case  the  fundamental  for- 
mulas of  thermodynamics  are  employed  in  a  new  field  for  their 
application,  and,  as  a  matter  of  fact,  the  entire  series  checks 
within  one  or  two  units  in  the  fifth  decimal  of  logarithms.  In 
the  solar  data  the  numbers  are  unusually  large,  but  even  the 
considerable  values  here  quoted  in  the  tables  are  not  sufficient 
to  reproduce  the  checks.  The  assumed  value  of  the  tempera- 
ture at  any  point  implies  the  succeeding  terms  as  computed. 


296  A  TREATISE   ON  THE   SUN*S  RADIATION 


f 
«o 
« 

00  OS  CO  -* 
CO  O  00  CD 

coSooos 

s 

d 

coS^oo             oSeo£         o 

d 

^t  t-ooos 

9 

O                                          r-i 

0 

§ 

^«  OS  ION 

1 

d 

coioc-oo             os  m  co  I-H         o 
d                        d 

d 

ioos  t-  os 
t-  c-oo  -^< 

d 

|9  <P  Of  CO 

«OOU3Tt 

M 

CO 

CD 

H 

c- 

t-t-eoo             oscot-oo         o         T-I 

00 

OOOSCftW 

tS 

0 

CO«Ot-00                   OST^NiH             O 

o 

Tt  t-OOOS 

rt 

o                      o 

0 

$ 

(M 

t-N«000 

OO  T}<  CO  t— 

H 

I 

1 

d 

co  ^  t-  so 

O**OOOS                   OTj<cO»-t             O 
•«*Wt-00                   OSTfCMi-l             O 

d 

ooost-o 
•^•ooosos 

K 

0                                0 

o 

fc 

0 

00 

?0 

t- 

00 

u 

£ 

d 

^SOOOOS        ^      W^NO             0 

d 

^ooosos 

M 

i 

d                o     °" 

en 

o 

8 

G 

w 

—   IS                                                           t-Tj<OS^) 

§     § 

3    1 

•*f 

CO 

U 

I 
d 

II                              co  m  <o  os 

!°,£g?s  *  ss?si    i^50^ 

U)Tt««D(»OS                   OOCO^-lO             0 

_o;  d                       d 

s 

1 

o' 

d 

M        i 

Q 

>                                                                                                             Tt<  T)<  00  N 

G" 

2       i 

| 
"So 

c 

3 
d 

ct                                                          moo  ^os 

^                                M                      10            S??kO<N 

1 

1 

d 

§^  OJ  t- 
oo  os  os 

o" 

««  d   "   '   '     3     o   '   '   ' 

>H                                                           (4 

-M 

M 

o 

i 

u 

"  1 

o 

s 

O                               0 
•**                                       m                                                 T-IOOO  rH 
-,                                           VH                                                COOS  COOO 

g                         O                            oososeo^ 

s 

1 

c 

1 

0 

'3    ^*  t—  00  OS          43         t*W^O           InO 

|d---  &  d---  v 

d 

IT- 

1 

c/5 

i 

rt                          <n                            «*H 

•s            °is§?s 

°  0^,0      °     comvoS     If*9! 

10 
OS 

s 

*°  wteo^n 

en 

d 

MOt-OSlO          W         OSU3r-lTj<         .go 
•g»Ot-OOOS        "g       «OOJrHO             O 

d 

^    ^C  00  CT5  ^ 

.So''       '.Sid'       '     .8   • 
o                         o                           73 

.Si  o 
o 

<: 

£                                {g                                      ||CO«ONCO 

€ 

H 

o                       o                         <-^cq  oo  »H  os 

o 

H 

0? 

®                                    ^                    CM  Ifi        '~lMMSo? 

c5 

(J 

00 

Ht^lO^^OO         II        OOCOW         ||C^C'3TTW 

OS 

||    ONt-00 

8 

d 

d 

-<  U5OOOSOS 

*-d               d  • 

0 

£ 

....              ....           .... 

H 

...           .... 

i 

, 

s 

i 

3 

I    '.         '                      ...             •    •    •    • 

1 

u 

*• 

'.    '.         '.                      '.'.'.             I    '.'.'. 

*• 

MM     MM   MM 

i  i  i  i 

COi-HO  1C 

d  d  o*  »-i            o*  d  o  1-4         o  d  o'  rn 

OOOi-( 

OTHER    SOLAR    PHENOMENA 


297 


50500  OJ 
>COrH  OO 
J«DOCO 


?3! 

d 


OSi-4tH< 

I-l  t-  rH  C 

moo  t-< 


oo  «>  o>  to 


gs£32 

.2  <N  (M  03  <0 

I" 


oot-  10 

coo.-i 

O5  i-H  CO 
W*0 


25! 
.S       ^^      SS! 


1C  00  OS  OS 


of 
1 


S^Sil    -S; 
I  d  '  '  '    & 


11   wos'S 


3.S.  i 

SSo 
eoTjH«> 

O*  O  O 


CJOOt-N  NOOt-C<I 

WTji^CJ  05-*!ON 

d  d  d  -«         do  o'  r-J 


298 


A  TREATISE   ON   THE   SUN  S   RADIATION 


0s!  CO  C^ 

OOOOOO 


~ 


T-H  CO  ^^ 

O5OiO5 


OiOOO5OcOcOOOOOl>OOO 

<N    CO    IO   Tfrl    CO   00   r-  1    <N   CO   CO   l^   rH    i—  1 


CO 
O 


CO 
OO 


O3OO<NOOOCOOOOiOO 

C^-l   tN*   00    00   O^   CD   CO    Oi   *O   00   i"H 

»OiOiOiO»OCDcOCO 


8 


3 
a 

o 
U 


N.  CO  00  CO  CO  O 
CO  C^  O5  O5  OO  Oi 
<N  d  rH  rH  rH  rH 


00  CO  I-H  CD  O5 
OO  I>  CO  -^  -^ 
(M  (M  <M  (N  <M 


CO  CO  CO  CO  (N  <N 


O  t>-  CO 

CO  CO 


00  **H  00  *O 

1>-  CO  O  O 

Tfl  ^  Tf  Tj< 


00  I>-  CO  ^O  >O  »O 
0 


C^  T-I  rt<  GO  W  O 

O5  O  (M  <M  rH  fH 

00  00  t>  CO  CO  CO 


I: 

CO  > 

o 


CO  CO  i—  i  i—  i 

co  *o  oo  co 

r^  ^  T^  10 


OTHER    SOLAR    PHENOMENA 


299 


««"  O  OS  3  TH  M 


t^>  03  OS  I***  CO  rH 


TjH    ^J    O 
CO  ^^  t^* 

to  cO  CO 


CO  l>  CO     .  *O  CO  1O  TH  CO  OS  00  O  O  00  *O 

J>-t>.lO    *O  l>  IO  CO  Tfl  rH  O  1>-  (NOOrH 

rH  cq  t^.   (N^fl>O  "*  00  CO  iO  O  00  O 

O      •  <N  Q     rH  CO  rH  rH  CO 


»O  CO  <N  »— i  W  CO 

i— I»O  COt>Oi»-t 

>OOS  i— (  T-H  ^  i-H 

i— i  CO  i— i  C^  CO  O5 


CO  CO  CO  ^jJJ       !>OOrHCO       (MOOTJHOO       COO5COI> 

O  O  C^OrHrHTtli— IC^COO 


CO   rH   O5 

(M  CO  00 

rH    C<l    CO 


(^rHi-HrH         (NOOO500         >OO5 
TtHCOCPOO         OSQOCOrH         COrH 


00        CO   »O  <M  rH 


rH   Q         r^   CO  rH 


(M  O  <N       »O  t»  CO  O5 
T-I  (M  iO       »-H  (M  ^  1-" 


OOCOCD  (NOOi-100  COCOi-HCO 

t^-COiO  i— iiOt^-^  i— (  i-H  O  >O 

<M   T}H   rH  ^I>(MT-I  I^COC^^ 

r—\  O           rHCO  Oi-HCl^O 


CO  00       OOO5C3CO       COC3COOO       O5<NOOCO 


>O  O  »O        t^  00  O 
CO  ^O  CO       "31  00  ^t1 


CO  l>  00  T^  to  (N  O 
00  -^  O  CO  (M  t^  rH 
Tfi  10  <N  OO  ^  00  CO 


(M  rH  |> 

CO  00  O5 

co  >o  -^ 


00       rH  t^»  C^  *O       O^l  CO  C^  t^* 

CO       OS  to  t^  C^       C^J  CO  CO  C^l 

TjH         OrH(Mt>.         (NWCOCO 


3.3.3.3.        3.3.3.3.        3.3.3,3.        3.3.3.3. 

CO»-tO>O        CO  i—  I  O  lO       COT-iOtO       CO  i—  I  O  »O 
C^OOt^fM       (NOOt^(N       (NOOt^lN       <NOOI^<M 


^n  »- 


f 

ffi 


3  ^ 

I  • 

U  S 


3    „ 


300  A   TREATISE  ON  THE   SUN'S  RADIATION 

No  attempt  has  been  made  to  modify  the  standard  formulas 
beyond  adapting  them  to  non-adiabatic  strata.  Such  further 
changes  as  may  be  required  by  other  physical  laws  must  be  a 
matter  for  further  research.  An  effort  will  be  made  to  dis- 
cover a  reliable  method  of  procedure,  which  can  be  utilized 
for  the  complex  molecular  systems,  as  well  as  for  the  monatomic 
gases.  It  is  probable  that  an  expert  application  of  the  spec- 
troscopic  physics  to  such  thermodynamic  data  will  be  required 
for  its  successful  completion.  The  practical  ways  to  obtain 
valuable  economic  forecasts  will  depend  upon  the  selection  of 
such  sensitive  variations  in  some  of  the  thermal  or  visible  lines 
of  the  spectrum  as  will  register  the  internal  solar  movements 
more  accurately  than  the  sun-spots,  faculae,  and  prominences 
can  ever  do.  It  is  evident  that  so  large  a  field  of  discovery 
has  been  opened  up  as  to  require  much  intelligent  cooperation. 

The  Variation  of  the  Intensity  of  the  Sun's  Radiation  in  Longitude 
as  Developed  by  the  26.68-Day  Synodic  Period 

Besides  the  question  regarding  the  variability  of  the  inten- 
sity of  the  sun's  radiation  as  a  whole  from  year  to  year,  as  in 
the  11-year  period,  there  remains  the  important  problem  of  the 
variability  from  one  meridian  of  the  sun  to  another,. as  has 
been  indicated.  The  variations  in  the  intensity  of  the  terres- 
trial magnetic  field  have  produced  the  26.68-day  period,  as 
corresponding  with  one  synodic  period  at  the  solar  equator, 
and  the  ephemeris  on  page  334  of  the  Meteorological  Treatise. 
It  is  obvious  that  the  practise  of  publishing  the  meteorological 
data  in  calendar  months  is  fatal  to  these  studies  in  solar  physics, 
because  it  obliterates  all  minor  periodic  effects.  This  unscien- 
tific method  should  be  abandoned. 

During  the  years  1912  to  1916  inclusive,  the  pyrheliometric 
observations  were  carried  on  regularly  at  Cordoba  and  Pilar, 
and  during  such  intervals  at  La  Quiaca  as  were  practical  through- 
out the  same  years.  The  observations  have  all  been  recom- 
puted by  the  method  described  in  Chapter  VI,  and  the  results 
were  collected  according  to  the  26.68-day  ephemeris.  It  turns 


OTHER    SOLAR   PHENOMENA  301 

out  that  the  series  at  La  Quiaca  has  seven  observations  on  the 
average  for  each  day  of  the  period,  and  the  Cordoba-Pilar 
series  an  average  of  fifteen  observations  on  each  day.  The 
curves  in  hand  represent  these  results  for  La  Quiaca,  3465 
meters  above  the  sea-level;  for  Cordoba,  438  meters,  and  for 
Pilar,  380  meters.  Similarly,  collections  of  other  data  follow: 
the  amplitudes  of  the  horizontal  magnetic  force  at  Pilar  1910- 
1916;  the  maximum  temperatures  at  Pilar  for  the  same  interval; 
the  relative  frequency  of  the  wind  direction  from  the  south, 
the  complement  being  from  the  north;  and  the  precipitation. 
These  data  are  designed  to  show  that  there  is  a  variation  in  the 
intensity  of  the  sun's  radiation  in  longitude  which  is  systematic 
and  persistent.  There  are  a  principal  maximum  near  the  15th 
day,  a  secondary  maximum  near  the  2d  day,  and  corresponding 
minima  near  the  8th  and  the  21st  days,  direct  for  the  meteor- 
ological data  and  inverse  for  the  radiation.  These  are  shown 
by  the  mean  curves.  There  are  eight  minor  maxima  on  each 
curve,  and  the  position  of  them  is  quite  harmonious  throughout 
the  series.  The  mean  synodic  variation  in  the  solar  radiation 
is  about  three  per  cent,  and  it  is  better  defined  at  the  high  level 
station  at  La  Quiaca  than  it  is  at  Cordoba-Pilar.  It  takes  a  large 
amount  of  data  to  separate  this  small  impulse  from  the  local  con- 
ditions. The  dust-effect  is  very  troublesome  to  eliminate  from  the 
radiation,  the  magnetic  storms  must  be  removed  from  the  ampli- 
tudes, and  the  general  circulation  disturbs  the  direct  effect  of  the 
solar  impulse  on  the  meteorological  elements.  The  primary  fact 
is,  however,  clear  that  the  sun  is  so  constituted  as  to  maintain 
two  axes  of  maximum  and  minimum  intensity  of  radiation 
at  right  angles  to  each  other,  and  two  minor  maxima  near  the 
extremity  of  each  axis.  These  results  are  in  harmony  with  those 
published  in  1895  and  1898,  so  that  the  26.68-day  period  is 
now  proved  to  have  maintained  itself  without  shifting  from 
1840  to  1916.  These  minor  crests  of  radiation  have  their 
effects  upon  the  circulation  of  the  earth's  atmosphere  and  the 
prevailing  weather  conditions.  It  follows  that  forecasts  of  such 
conditions  can  be  made  by  projecting  the  fundamental  curve 
according  to  the  ephemeris,  and  experience  shows  that  it  is  of 


302  A  TREATISE  ON  THE  SUN'S  RADIATION 

practical  value  for  long-range  forecast  conditions.  There  is 
no  evidence  that  a  25-day  period,  and  an  18-day  lag  in  the 
effect,  has  any  validity;  on  the  other  hand,  persistent  maximum 
impulses  on  the  26.68-day  period  certainly  prevail.  There  is 
no  doubt,  also,  that  an  equipment  suitable  for  following  the 
solar  action  closely  will  make  it  possible  to  interpret  many 
of  the  seeming  discordances  in  terms  of  profitable,  scientific 
forecasts  of  the  weather  to  be  expected. 

These  data  and  opinions  contradict  many  of  the  ideas  at 
present  prevailing  regarding  this  matter.  Meteorologists  have 
generally  failed  to  obtain  good  results  because  they  do  not 
employ  a  sufficient  amount  of  homogeneous  data,  they  do  not 
utilize  the  26.68-day  period  in  their  compilations,  nor  do  they 
recognize  the  phenomena  of  the  double  annual  inversion  in 
the  effects  under  local  conditions.  Failing  these  essentials, 
their  views  must  be  disregarded.  Solar  spectroscopists  note  the 
variable  frequency  in  the  sun-spots,  the  faculae,  the  flocculi, 
and  the  prominences,  and  it  has  been  proved  that  they  syn- 
chronize annually  with  the  magnetic  field  and  the  intensity  of 
the  radiation.  We  have  now  proved  that  the  intensity  of  the 
radiation,  the  amplitudes  of  the  magnetic  and  the  meteorolog-, 
ical  fields  synchronize  in  the  26.68-day  rotation,  and  the  infer- 
ence must  follow  that  the  visible  markings  on  the  sun  neces- 
sarily have  definite  relations  to  the  maxima,  which  have  their 
positions  in  fixed  solar  longitudes.  It  should  be  remembered 
that  the  spectroheliograph  is  effective  in  the  superficial  layers 
of  the  solar  gases,  generally  near  the  top  of  the  sun's  isothermal 
layer,  while  the  radiation  is  generated  primarily  near  the  bottom 
of  the  isothermal  layer  of  each  constituent  gas.  While  the  surface 
phenomena  must  have  definite  relations  to  the  low-level  im- 
pulses, it  is  evident  that  the  circumstances  seen  in  the  lines 
of  the  spectra  are  not  entirely  comprehensive.  At  present,  it 
is  required  to  trace  out,  so  far  as  possible,  the  relations  between 
the  variations  in  the  spectral  lines,  the  surface  solar  conditions, 
and  their  connection  with  the  original  radiation  processes. 
Similar  remarks  apply  to  the  spectrobolometer. 

Furthermore,  it  appears  from  the  computations  that  some 


OTHER    SOLAR   PHENOMENA  303 

of  the  adopted  theories  regarding  the  Wien-Planck  and  the 
Stefan  formulas  of  radiation  need  modification,  and  it  would 
be  easy  to  point  out  several  of  the  details  of  such  a  reconstruc- 
tion. This  work  is  being  prosecuted  as  rapidly  as  possible,  in 
view  of  the  complexity  of  the  fundamental  facts  of  the  structure 
of  matter.  It  is  mentioned  in  this  place  only  to  emphasize 
the  opinion  that  the  subject  has  been  by  no  means  exhausted, 
and  that  meteorologists  and  astrophysicists  should  generally 
enter  upon  the  new  methods  of  Discussion  which  have  been 
opened. 

Especially,  it  will  be  necessary  to  set  apart  a  few  meteorolog- 
ical stations  and  astrophysical  observatories  for  this  purpose. 
It  is  supposed  that  fifty  meteorological  stations  and  six  solar 
physics  observatories  "will  be  sufficient  under  an  international 
supervision,  to  follow  the  solar-terrestrial  synchronism  practi- 
cally, and  to  make  progress  in  the  necessary  physical  studies. 
The  economic  value  should  be  considered  from  the  world-view, 
rather  than  from  that  of  any  nation.  Such  long-range  fore- 
casts annually  are  already  possible  in  certain  localities,  such  as 
Argentina,  where  conditions  are  favorable.  It  would  be  very 
rash  to  attempt  to  set  any  limit  to  the  possible  development  of 
a  function  already  proved  to  be  world-wide  and  to  persist  for 
more  than  half  a  century,  especially  now  that  it  has  become 
possible  to  determine  the  26.68-day  variability  of  the  intensity 
of  the  solar  radiation  by  means  of  simple  pyrheliometers  when 
the  computations  are  made  in  a  definitive  form. 

The  method  of  computation  of  the  intensity  of  the  solar 
radiation  employed  in  this  Treatise,  makes  the  pyrheliometer 
independent  of  the  complex  bolometer,  and  this  simplification 
of  the  procedure  renders  it  more  practicable  to  extend  the 
observations  to  many  cooperative  stations.  Simple  working 
instructions  can  be  readily  prepared  for  general  use. 


CHAPTER  VIII 

Three  Theories  of  Radiation 

In  the  following  notes  a  brief  account  will  be  given  of  the 
three  theories  of  the  origin  of  electromagnetic  radiation,  which 
seem  to  suggest  further  research  and  verification. 

1.  Planck's    Theory    of   Oscillators.    The    atoms    are    con- 
ceived to  have  polarized  charges  on  their  spherical  surfaces, 
which  oscillate  in  periodic  frequencies  equal  to  those  of  light. 
The  form  of  the  Wien-Planck  function  for  the  intensities  in  the 
spectrum  was  deduced  from  the  theorems  of  probability  and 
entropy.    It  seems  to  be  well  verified  by  experiments,  at  least  as 
to  its  form.     The  conditions  imposed  upon  the  radiation  were 
strictly  adiabatic,  the  coefficients  k,  h  are  constants,  but  these 
are   not   applicable   in    free   atmospheres   except   in   specified 
regions. 

2.  Bigelow's  Theory  of  Collisions.    The  Planck  formula  has 
been  reproduced  as  to  its  form  from  the  non-adiabatic  thermo- 
dynamics, wherein  k,  h  are  wide,  variable  coefficients,  the  periodic 
frequencies  being  derived  from  the  collisions  reacting  upon  the 
electrons  of  a  nucleus-sphere.     The  negative  charges  circulate 
in   orbits,    arranged   upon   spherical   surfaces,    which   undergo 
vibratory  oscillations  as  the  result  of  the  shocks  from  the  col- 
lision of  such  dynamic  systems. 

3.  Bohr's  Theory  of  Non-Radiating  Orbits.  The  Rutherford 
positive  nucleus  controls  a  series  of  coplanar  orbits  of  negative 
electrons  circulating  in  rings  of  stability  which  are  non-radiative. 
The  periodic  frequency  is  due  to  the  electrons  passing  from 
one  orbit  to  another,  as  the  result  of  changing  the  configurations, 
and  the  radiation  is  the  loss  of  energy  during  the  readjustments 
of  the  circulating  electrons. 

304 


PLANCK'S  THEORY  OF  RADIATION  305 

It  seems  proper  to  present  briefly  the  most  important  features 
of  these  three  theories,  for  the  sake  of  facilitating  their  further 
discussion  and  improvement,  because  it  is  not  probable  that  the 
structure  of  the  atom  is  at  present  sufficiently  understood. 

The  Derivation  of  the  Wien-Planck  Formula  for  Black  Radiation 

in  a  Spectrum 

The  primary  question  in  this  branch  of  physics  is  whether 
thermodynamic  conditions  in  the  gases  are  wholly  consistent 
with  the  electromagnetic  radiation,  or  whether  some  unknown 
dynamical  term  is  required  to  bridge  the  gap.  On  the  adiabatic 
basis  of  k,  h  as  constants,  it  seems  impractical  to  solve  this 
problem,  but  on  the  basis  of  non-adiabatic  variable  coefficients 
the  subject  enters  upon  a  new  phase  of  discussion. 

The  Wien-Planck  formula  for  black  radiation  in  the  spec- 
trum is  based  upon  an  analytic  discussion  of  the  Theories  of 
Probability,  Equipartition  of  Energy,  Entropy,  and  Thermo- 
dynamic Energy.  It  is  desirable  to  make  the  basis  as  specific 
as  possible,  especially  in  view  of  the  fact  that  k  and  h  are  not 
constants.  The  following  statement  somewhat  modifies  that 
followed  by  Planck,*  Richardson,!  and  others. 

Let  k  T  represent  the  kinetic  energy  of  an  electron ;  let  h  v 
represent  the  potential  energy  of  an  electron,  where  T  is  the 
temperature  of  the  medium  and  v  is  the  frequency  of  oscillation, 
probably  between  separate  electrons,  rather  than  the  oscillation 
of  polar  charges.  Inner  energy  of  one  electron  U  =  k  T  +  h  v. 

(217)  Let  77  =  -~  =  ,      _^        =  the  probability  of 

radiation  occurring. 


_  . 

rj        h'v  hv  hv 

1  1  7>  T* 

(219)  = 1  =  -: —  =  p  .  uv  proportional  to  the 

f\  rj  h  v 

radiation  density  at  the  v  frequency. 

*  Planck,  "  Warmestrahlung,"  p.  139. 

t  Richardson,  "  Electron  Theory  of  Matter,"  p.  353. 


306  A   TREATISE   ON   THE    SUN'S    RADIATION 


_K  +  !._*_!  +  A      SL    !.-*T,l 

hv^~  2   "  hv  "  2'         A,        2   "   hv  "  2* 

(222)  From  the  Boltzmann  Entropy  Law,  S  =  k  log  w,  Planck 
deduces  the  probability  formula, 


(223)  NS  =  k\ogW=-Nk 

o 

Take  P  =  —  ,  and  give  m  successive  values, 
"n 

(224)  NS  =  -Nk  [-^log  „  +  (|  -  l)  log  (~  -  l)]. 


1        U        I 

Substitute  —  =  7  --  h  IT,  and  differentiate  for  dS,  d  U. 

t]         n  v        Zi 


U  J_  £_T      _3_ 

J     O                   1                      /,                     7  l         O  Z.                      /.             I         O 

(io             1              /?             tl  V  £  R              It  V            £ 

T~"  o"  "TT  +  "o" 

n  V  £  ft  V            £ 


£  .      /         ^^\       .^.  1  //z^V  .    X   (hvY 

rv\og  (I  +  j-j,  )  omitting  —  j  ^  +-3   V^rJ  ""'  ' 

(227)  Hence,       .  =  log  (l  +  ^),  and  «H  =  i  +  ^. 


^  T  1 

(228)  The  radiation  p  uv  =  T~  =  -*r  -  ,  for  the  ^  frequency. 


Integrating  for  the  total  spectrum, 

STT^  /Vrfv         487r/f 


PLANCK'S  THEORY  OF  RADIATION  307 


(230)  u  =        -     ~  T*  =  a  T4,  Stefan  Law.  ' 
c         n 

These  formulas  are  first  approximations  to  natural  conditions 
of  radiation,  and  since  k,  h,  a,  are  not  universal  constants,  they 
are  of  only  formal  value,  though  they  can  be  used  in  succession 
wherever  a  certain  set  of  values  (k,  h)  are  adopted.  They, 
therefore,  lead  to  families  of  spectra,  which  are  superposed  on 
the  same  level  in  atmospheres  consisting  of  several  independent 


gases.     The  formula  of  integration  (227)  T-~,  =  log  ( 1  +  r 

K  J.  ^  K 

is  tautological  when  ryris  a  small  term;    generally  it  is  not 


decisive.  The  following  computations  show  how  far  the 
adiabatic  case,  upon  which  Planck's  discussion  is  founded, 
is  from  satisfying  the  natural  conditions  of  atmospheric 
radiation. 

Planck's  function  is  a  form  of  the  probability  curve,  which 
gives  the  distribution  of  the  ordinates  of  intensity  according  to 
their  relative  frequency  of  occurrence  in  seeking  a  maximum 
parameter,  the  Temperature.  It  seems,  therefore,  that  the 
thermodynamic  conditions  in  a  given  volume  of  an  atmosphere 
should  be  capable  of  reaching  the  same  result  from  the  col- 
lisions of  n  molecules.  We  shall  proceed  to  deduce  two  formulas, 
one  for  the  non-adiabatic  branch  and  one  for  the  adiabatic 
branch  of  the  curve  for  h,  into  which  the  Planck"  formula  divides 
itself  at  the  bottom  of  the  solar  isothermal  layers,  as  heretofore 
explained.  Compare  Fig.  32  for  these  three  curves  of  the 
potential  h  of  the  vibrating  electrons. 

Derivation  of  Bigelow's  functions  for  the  potential  h.  The  first 
of  the  formulas  is  for  the  non-adiabatic  branch,  and  it  is  derived 
from  the  terms  P.p.R.T,  taking  them  in  the  bulk,  as  an  integrated 
effect.  The  second  of  the  formulas  is  for  the  adiabatic  branch, 
and  it  is  deduced  directly  from  the  constituents  of  P.p.R.T, 
by  the  derivatives  in  terms  of  n.  N.  T,  which  are  much  more 
fundamental.  Indeed,  it  will  be  proved  that  (n.  N.  T)  is  a 
better  system  for  thermodynamics  than  the  usual  (P.  p.  R.  T). 


308  A   TREATISE   ON    THE    SUN'S    RADIATION 

The  General  Equations  of  Condition 


d  P         n  K      d  T          K 

-J-  =  7=l--T+^ 


i)^-"^  log  (). 

pO/  K   -  \1  O/ 

dp          n     dT  1  /T\ 

—  =  --  7  -Tfr  +  -  r  log  (  _  )  d  n. 

p  K   —    IT  K   —    I  \./o/ 

iog 


The  Boyle-Gay  Lussac  Law,  with  all  terms  variable. 

(234)  Pv  =  RT.     Differentiate  and  make  the  substitutions. 

(235)  Pdv  +  vdP  =  RdT  +  TdR. 


J 

K  —   1  K  — 

(238)  RdT  ........................  =  RdT. 

r—  ^7    T^  T^ 

(239)  T  d  R  =  T     #  (n  -  1)  -r  +  ^  log 


(»  -  1)  R  d  T  +  R  T  log  dn 


(240) 

This  is  the  non-adiabatic  form.     For  n  =  1,  we  have, 
(241)  Pdv  +  vdP  =  RadT,  the  adiabatic  form. 


BIGELOW'S    FIRST    THEORY   OF    RADIATION  309 

7    rrt 

Since  n  =  ~r£>  and  g  d  z  =  —  Cpa  d  Ta,  we  have  by  sub- 
stitution and  addition, 

(242)  Pdv  +  vdP  +  gdz  =  -(Cpa-  R)dTa  +  RT  log 

T 


This  is  the  equation  of  condition  in  non-adiabatic  atmos- 
pheres.    Hence,  by  integration,  and  with  P  d  v  =  d  W, 

(243)  gio  (*i  -  20)  =  -  Pl~P°  -  (Cpa  -  Rw)  (Ta  -  To)  - 

Pio 

(Wi  -  W0)  +  #10  rw  log  (Y\   (n^  -  no). 

(244)  gio  (zi  —  2o)  =  — 


PlO 

Rw  r,0  log 


In  my  computations  the  last  term  in  (HI  —  nQ)  has  been 
omitted,  but  more  exactly  it  must  be  included.  My  equations 

for  radiation  depend  upon  —,  —  =  K,  while  Planck's  depend 

upon  T7T~\  =  J-     It  is  interesting  to  note  that  they  have  such 

close  relations  as  have  been  already  pointed  out,  besides  others, 
as  in  the  examples  on  the  following  pages. 

It  is  evident  that  the  Wien  displacement  law  Xm  T  =  0.2891 
constant  applies  only  to  the  adiabatic  case,  while  Xm  T  is  a 
variable  in  all  non-adiabatic  atmospheres.  It  is  thought  that 
these  values  are  too  small,  according  to  certain  bolometric 
data,  and  that  the  Xm  T  values  for  XOT  in  centimeters  may  have 
to  be  decreased  in  the  logarithm  by  about  0.30000  to  make  the 
Xm  somewhat  larger.  This  amounts  to  moving  the  Planck 
plotted  curve  a  little  to  the  left  of  its  position.  This  discrepancy 
may  be  due  to  imperfections  in  the  computation,  such  as  the 

rr\ 

omission  of  the  term  R  T  log  (jr  J  d  n  in  the  equation  of  con- 

dition.    Until  further  progress  has  been  made  in  these  details 
the  subject  must  be  left  open  for  study. 


310  A   TREATISE   ON    THE    SUN'S    RADIATION 

Additional  Formulas 

It  is  seen  that  the  derivatives  are  only  for  P.  p  .  R  .  T,  which 
treat  of  the  gas  from  the  integrated  or  the  bulk  conditions, 
rather  than  from  their  individual  molecular  constituents. 

The  full  significance  of  the  change  of  the  adiabatic  constants 
to  the  non-adiabatic  variable  coefficients  in  atmospheres  cannot 
be  completely  anticipated,  especially  in  molecular-atomic-elec- 
tronic physics.  However,  there  emerge  from  the  preceding 
computations  certain  general  formulas,  of  which  a  few  may  be 
mentioned. 

Cp       5 

Kinetic  energy  in  monatomic  gases,  K  =  -^-  =  —  . 


(245)      - 

o 

NkT  =  - 


Cp  .  5 

Kinetic  energy  in  gases  whose  K  =  g-  is  not  — 


(246) 


. 

—   13  K  —   1  K  —   1  K  — 

N  k  T  =  ——r     -kTV  =  HV  =  CvPTV 


K 


Specific  heat  at  constant  volume  Cv, 

1        PV          1      NkT      HV          1      11 

(247)   Cv  =  -  -  --  ™  =  -  -  --  Tfr-  =  —  ™  =  -  T  T-  q2  ™. 
K  —  1    mT      K  —  1   mT        m  T      K  —  1  3  *    T 


The  specific  heat  in  terms  of  k,  h,  v  (Wien-Planck), 

h  V 

_h2v2       e"T 


e"T- 


»  (Einstein). 


The  inner  energy  in  the  kinetic  theory  of  gases  is  related  to 
the  pressure  as  follows: 


(249)  (U,  -  U0)K  =        -j  (Pi  -  P.)  =  rf  (A  -  P.)     ' 

by  the  kinetic  theory. 

(250)  If  (Ui  -  U0)Th  =  C»w  (Ta  -  T0)Th  =  (Cpa  -  R1Q)  (Ta  -  T0) 


BIGELOW'S    FIRST   THEORY   OF    RADIATION  311 

is  the  inner  energy  as  derived  from  the  non-adiabatic  thermo- 
dynamics; the  two  are  connected  by  the  following  formula: 


(251)   (Ui  -  U0)K  =  P10       -£-T 


g   *  - 


Pressure,  number  of  molecules,  and  the  kinetic  energy. 

(252)  P  =  nk  T  =  4-  P  ?2-       PV  =  NkT  =  -^-mq2. 

o  o 

(253)  -|  *  T  =  -i-  •  y  P  ?2  =  i  •  y  m  $»,  for  three  degrees 

of  freedom. 


(255)  Po  -  Pi  =  Wio  k  T  -rf  log  -  =  c  (zi  -  2»)  ^  Pio  Tw  h. 

JV1  HI  & 

It  follows  that  Planck's  Wirkungsquantum  h  can  be  expressed 
in  thermodynamic  terms. 

(256)  h= 


-  So)    _w     Tio         Pio 

2 

c  is  the  velocity  of  light,  (zi  —  zb)  the  depth  of  the  column 
of  gas  where  P0  is  the  pfessure  at  ZQ,  and  PI  that  at  z\\  Tw  is  the 
mean  temperature  of  the  column  in  which  the  gradient  has  the 

wi 
same  value  through  the  stratum;  —  is  half  the  atomic  weight  in 

Zi 

monatomic  gases,  and  it,  therefore,  is  the  number  of  the  electrons 
in  each  molecule  of  this  kind.  Hence,  h  represents  the  potential 
energy  which  exists  between  pairs  of  electrons  per  degree  in  the 
simplest  form  of  the  gaseous  structure.  This  seems  to  be 
fundamentally  true  in  solar  gases,  but  to  need  an  additional 
term  in  terrestrial  gases. 

It  was  proved  by  trial  computations  that  —  must  be  used 


312  A    TREATISE    ON   THE    SUN'S    RADIATION 

and  not  m,  as  if  about  one-half  of  the  number  of  electrons  form- 
ing the  atom  were  concerned  in  this  potential  action.  Since 
the  adiabatic  branch  has  a  different  function  it  seems  to  be 
implied  that  the  electrons  may  be  in  another  form  of  configura- 
tion, perhaps  wholly  dissociated  under  the  very  high  prevailing 
temperature,  more  than  8000°.  It  should  be  noted  that  in  the 
Planck  formula,  wherein  c  .  h  .  k  are  assumed  to  be  universal 

constants  in  the  exponent,  ,     /r,  such  that  for  Cz  =  —r  =  con- 

K  \  J.  K 

stant.     For  the  adiabatic  case,  the  distribution  is  different  in  the 

non-adiabatic  case.     Here  L  .   ^  becomes  T-^F  and  k  T  is  a  cou- 

rt \  1  k  1 

stant  for  each  atmosphere  under  its  gravitation,  while  h  and  v  are 
both  variables.  It  is  entirely  probable  that  the  potential  h 
between  electrons  of  the  same  gas  should  vary  in  value  from 
one  level  to  the  other  while  the  kinetic  energy  is  a  constant, 

3  1 

—  k  T  =  —  mu2  =  constant  for  each  atom. 

£  £ 

The  Variable  k  in  All  Atmospheres 

It  has  been  verified  throughout  the  computations  that 
(k  T)E  has  one  constant  value  in  the  earth's  atmosphere,  and 
that  (k  T)s  has  a  constant  value  in  the  sun's  atmosphere,  such 
that, 

(257)  (k  T)E  X  72  =  (£  T)s.  log  T2  =  2.89514. 

C.  G.  S.        M.  K.  S. 

(258)  Earth,  (kT)E  =  3.7145  X  10-  14   -14.56990     -21.56990 

(259)  Sun,      (kT)s  =  2.9179  X10-11   -11.46508     -18.46508 

The  kinetic  energy  of  each  atom  is  one  constant  in  the 
earth's  atmosphere  and  another  constant  in  the  sun's  atmosphere. 
They  are  so  related  that 


(260)         r  =  T*,  where  T  =  —  =  28-028> 
(*T)B  g 

which  is  the  ratio  of  the  surface  gravity  accelerations. 

2 

This  is  correct  since  we  assumed  k  T  =  —  E0  =  constant. 

o 


BIGELOW'S   FIRST   THEORY   OF    RADIATION  313 

The  Atmospheric  Pressure 

Since  P  =  n  k  T,  it  follows  that  the  pressure  in  gases  depends 
upon  the  number  of  molecules  present  in  the  unit  volume. 
Furthermore, 

(261)  P  =  n  k  T pressure, 

(262)  p  =  n-^      density, 

N 

(263)  R  =  —  k     the  gas  efficiency,  and 

(264)  Boyle-Gay  Lussac  Law,P  =  PRT  =  nkT  =~ .  ^  .k  T. 

These  propositions  greatly  simplify  the  theory  of  gases, 
especially  in  view  of  the  probable  numerical  number  of  the 

m 
electrons  =  — . 

Zi 

The  Variable  Potential  Coefficient  h 

It  remains  to  illustrate  the  relation  of  the  variable  potential 
coefficient  to  the  computed  volume  density  of  the  radiation 

by  the  two  formulas: 

*4  1  IIP   P 

(265)  Thermodynamic,  hB=  —T-     — r  .  -  .  •=-  .  -^——,  -  -  > 

c  (Zi  —  ZQ)     m     1 10         r  10 

2 

Bigelow. 
/48  TT  R  \ 1/8  k'13 

(266)  Radiation,  hP=  ( -}    — , Planck. 

^     a     '      c 

In  tables  98-102  is  collected  the  summary  of  the  computed 
values  of  u,  using  these  two  values  of  h  for  the  sun  and  the 
earth,  respectively.  They  are  collected  by  temperatures,  instead 
of  by  heights,  since  the  several  gases  give  nearly  the  same  values 
of  k,  h,  at  the  same  temperatures,  as  computed.  This  applies  to 
the  monatomic  elements  and  hydrogen  in  the  Bigelow  formula, 
but  hydrogen  is  given  separately  under  the  Planck  formula. 

In  the  Bigelow  computation  the  exponent  a  is  taken  4.00, 
as  in  the  Stefan  formula,  while  in  the  Planck  formula  both 
h  and  a  vary  in  such  a  way  as  to  produce  very  nearly  the  same 
value  of  the  volume  density  of  the  radiation  u,  as  can  be  seen 


314 


A   TREATISE    ON    THE    SUN'S    RADIATION 


by  comparing  Tables  98  and  99.  This  is  an  additional  proof  that 
the  solar  radiation  is  black,  and  it  has  been  hereby  computed 
directly  from  the  thermodynamic  non-adiabatic  data. 

In  the  case  of  the  earth's  atmosphere,  there  is  brought  out  a 
large  discrepancy  in  the  Bigelow  formula,  as  in  Tables  100  and 
101.  The  hB  (Bigelow)  is  much  smaller  than  hp  (Planck),  and  the 
resulting  u  is  impossible.  On  computing  the  correction  AA,  which 
is  required  to  change  h  into  hi  =  h  -\-  Ah,  such  that  u\  =  u  in 
Tables  100  and  101,  the  values  of  Ah  are  seen  in  Table  101. 

In  order  to  exhibit  these  relations,  they  are  plotted  in  Fig.  32. 
The  hB  formula  extends  from  -24.60000  to  -20.00000  in  a 
smooth  curve;  the  hp  extends  from  -26.60000  to  -26.00000, 
thence  by  a  sudden  sal  turn  to  —23.10000,  whence  it  follows 
nearly  along  the  hB  curve.  The  saltum  occurs,  as  already 
indicated,  near  the  bottom  of  the  sun's  isothermal  layer,  and 
it  is  apparently  related  to  the  change  of  configuration  from 
free  electrons  into  atoms  or  monatomic  molecules.  The  dia- 
tomic H2  is  intermediate. 

TABLE  98 
THE  SOLAR  VOLUME  DENSITY  OF  RADIATION 

*i*-a7v. 


Using 

7lfc                            i        i      i-o  -  -n  rB-     , 

w) 

'lB  ~   ^  („           »\    '    ~»         T      *           P             \.£>lgClC 
C  \Z\  —  ZQ)        7n        JL  10             •*!() 

~2~ 

T 

k 

h 

a 

u 

679.. 

Log. 
-14.65022 
-14.34924 
-14.10323 
-15.94234 
-15.82945 
-15.74361 
-15.66212 
-15.58445 

-15.59363 
-15.56757 
-15.51222 
-15.46630 
-15.42037 
-15.38579 
-15.34803 

Log. 
-22.94069 
-22.46783 
-22.07325 
-23.79901 
-23.60079 
-23.46848 
-23.25572 
-23.13332 

-23.12281 
-23.01921 
-24.94727 
-24.85549 
-24.78105 
-24.69859 
-24.63058 

4.000 
4.000 
4.000 
4.000 
4.000 
4.000 
4.000 
4.000 

4.000 
4.000 
4.000 
4.000 
4.000 
4.000 
4.000 

Log. 
-11.88767 
-  7.46057 
-  6.55587 
-  6.43543 
-  5.93421 
-  5.32002 
-  4.94194 
-  3.24510 

-  3.33051 
-  3.68471 
-  3.88839 
-  2.29867 
-  2.37027 
-  2.62641 
-  2.79080 

1484  

2485  

3719  

4564 

5526  
6604  
7629  

7687  

8369  

9443  

10511  

11552  

12572  
13405  

BIGELOW'S   FIRST   THEORY   OF    RADIATION 

TABLE  99 

/487r/3\M  &4/3  ,_. 
Using  &p  =   ( I      —      (Planck) 

For  the  monatomic  elements 


315 


r 

k 

h 

a 

u 

679.. 
1484  
2485  . 

-22.83036 
-22.23233 
—23  83910 

3.7556 
3.9103 
3  9701 

-9.52655 
-7.88278 
—5.15670 

3719  
4564  

-23.60358 
-23.45863 

3.9860 
3  .  9923 

-5.97200 
-4.33229 

5526  
6604. 

-23.34590 
-23  22775 

3.9975 
3  9836 

-4.67842 
—4  97341 

7629.  . 

Same 

—23  09578 

4  0179 

—3.42749 

7687.. 
8369  
9443  

values 
of  k 

-26.92865 
-26.88846 
-26.81290 

2.4100 
2.4185 
2.4182 

-3.73345 
-3.87328 
-3.96063 

10511..  
11552  
12572.. 

-26.75073 
-26.66880 
—26  63955 

2.4222 
2.4189 
2  4217 

-2.26759 
-2.27328 
-2.33339 

13405  

—26  59442 

2  4191 

-2.36464 

For  the  diatomic  hydrogen 


T 

k 

h 

a 

u 

7687.. 
8369. 

Same 

-24.06865 
—24  00975 

3.3316 
3  3468 

-3.89554 
—2  15091 

9443  
10511  

values 
of  k 

-25.94549 
-25  88354 

3.3483 
3  3490 

-2.04349 
-2.59640 

11552  

—25  82674 

3  .  3487 

-2.59076 

TABLE  100 

THE  TERRESTRIAL  VOLUME  DENSITY  OF  RADIATION 

487T/3     fr     , 
c*      '  h3  ' 
1  J_     J_     Po-Pi 

L  -  zo)  '  m_  *  Tio  ' 
2 


u  = 


a  T°-. 


Using  hB 


(Bigelow) 


z 

T 

k 

h 

a 

u 

Meters 

Log. 

Log. 

Log. 

66000... 

52.0 

-15.03573 

-20.99536 

4.000 

-26.80222 

56000... 

86.7 

-16.71084 

-19.14377 

4.000 

-27.94551 

46000.  .  . 

109.3 

-16.55127 

-20.83233 

4.000 

-26.64395 

36000... 

216.3 

-16.23881 

-20.22718 

4.000 

-24.37532 

26000... 

222.5 

-16.22207 

-20.19276 

4.000 

-24.48070 

16000... 

217.3 

-16.23216 

-20.22384 

4.000 

-24.38674 

6000... 

257.0 

-16.16597 

-20.06449 

4.000 

-24.89151 

2000.  .  . 

279.8 

-16.12525 

-21.99349 

4.000 

-23.08931 

500.  .  . 

285.5 

-16.11579 

-21.98890 

4.000 

-23.10028 

316 


A   TREATISE   ON   THE    SUN'S    RADIATION 


TABLE   101 


/48x£\M    £4/3 
Using  hP  =(  —  —  1       — 


._ 
(Planck). 


z 

T 

k 

h 

a 

u 

66000... 

52.0 

-25.47579 

2.737 

-11.19360 

56000... 

86.7 

-26.62427 

3.399 

-  8.33910 

46000... 

109.3 

-26.26206 

3.601 

-  7.54148 

36000... 

216.3 

Same 

-27.86323 

3.577 

-  6.49953 

26000.  .  . 

222.5 

values 

-27.77386 

3.668 

-  6.95808 

16000.  .  . 

217.3 

of* 

-27.71347 

3.768 

-  5.36961 

6000.  .  . 

257.0 

-27.58766 

3.819 

-  5.88588 

2000.  .  . 

279.8 

-27.52770 

3.826 

-  4.06088 

500... 

285.5 

-27.51076 

3.829 

-  4.11486 

TABLE   102 

Correcting  Bigelow's  Formula,  hi  =  h  +  A  k 


z 

r 

Ah 

hi 

a 

Ml 

66000... 

52.0 

-6.20287 

-24,19823 

4.000 

-11.19361 

56000... 

86.7 

-7.86880 

-25.01257 

4.000 

-  8.33911 

46000... 

109.3 

-7.70082 

-26.53315 

4.000 

-  7.54149 

36000... 

216.3 

-7.96526 

-26.19244 

4.000 

-  6.49954 

26000... 

222.5 

-7.84087 

-26.03363 

4.000 

-  6.95809 

16000... 

217.3 

-7.67271 

-27.89655 

4.000 

-  5.36861 

6000... 

257.0 

-7.66854 

-27.73303 

4.000 

-  5.88589 

2000... 

279.8 

-7.67614 

-27.66963 

4.000 

-  4.06089 

500.  .  . 

285.5 

-7.66175 

-27.65071 

4.000 

-  4.11485 

Extend  the  branch  of  the  hp  curve  to  —24.30000.  Since 
the  transformation  of  energy  is  accompanied  by  an  emission  of 
radiation,  and  an  accession  to  the  potential  energy,  we  may 
regard  the  area  between  hB  and  hp  as  a  measure  of  this  energy  of 
stored  potential  in  the  structure  of  the  atoms. 

In  the  earth's  atmosphere,  the  hp  curve  appears  at  the  top  of 
the  diagram,  —27.50000  to  —25.00000,  while  the  other  curve 
kB  is  near  the  vanishing  point  of  the  solar  curves  at  —21.00000. 
The  differences  between  hB  and  hp  are  given  in  Table  102.  It 
will  be  desirable  to  investigate  further  the  relations  between 
these  formulas. 


BIGELOW'S   FIRST   THEORY   OF    RADIATION 

THREE  THEORIES  OF  RADIATION. 


317 


log.  h  -20,00000    -21.00000    -22.00000    -23.00000     -24.00000     -25.0COOO    -26.00000    -27.00000 
The  variable  potential  energy  of  radiation  h. 

1  1  1         Po-Pi 


I,  Non-adiabatic,  hy  = 


(Zi  —  So)          W»         C 

2 


Hrtl           i                              i               /48  7T  /3\  '3     w    '• 
,  Planck,  hp  =  ( —   . 

\    a     J      c 

III,  Adiabatic,          HA  =  -  ^^"^  •  -  •  -• 

(Vi  —  Vo)i0      n       vm 


FIG.  32.     The  Non-Adiabatic  and  the  Adiabatic  Branches  of  the  Curve  of  the  Functions 

of  Solar  Radiation  h. 


318 


A    TREATISE   ON    THE    SUN'S    RADIATION 


Certain  Relations  Between  the  Adiabatic  and  the  Non-Adiabatic 

hv 
Values  of  y-y . 

It  is  proper  to  study  the  factors  which  are  required  for 
reducing  the  non-adiabatic  values  of  h  v/k  T  in  the  solar  and 
terrestrial  atmospheres  to  those  required  to  conform  to  the 
Planck  formula,  as  developed  for  the  variables  (k  .  a).  Table  103 
gives  the  factor  of  reduction  for  hB,  which  is  A  c  .  73. 


TABLE  103 
EVALUATION  OF  ( -^ 


I.    Terrestrial  Conditions. 


T 

hB 

hB*m 

log  h    v 

1      (kV\ 

Ac.y3 

og    m 

agAc.y* 

l°g\kT)B 

52  

-26.51601 

12.73207 

-13.24808 

-13.69795 

0.67818 

1.12805 

86.7  

-26.66442 

12.95409 

-13.61851 

-13.45265 

1.04861 

0.88275 

109.3  

—26.35298 

13  05469 

—13  40767 

—13  34496 

0  83777 

0  77506 

216  3  ... 

—27  74783 

13  35113 

—13  09896 

—13  20430 

0  52906 

0  63440 

222.5  

-27.71341 

13.36340 

-13.07681 

-13.17228 

0.50691 

0.60238 

217.3  

-27.74449 

13.35313 

-13.09762 

-13.09123 

0.52772 

0.52133 

257.0  

-27.58514 

13.42600 

-13.01114 

-14.99801 

0.44124 

0.42811 

279.8  

-27.51414 

13.46292 

-14.97706 

-14.99233 

0.40716 

0.42243 

285.5  

-27.50955 

13.47168 

-14.98123 

-14.98743 

0.41133 

0.41753 

II.    Solar  Conditions. 


273  

-21.50777 

13.45223 

-8.96000 

-8.75200 

3.49492 

3.28629 

679  

-22.94069 

13.84794 

-8.78863 

-8.59493 

3.32355 

3.12985 

1484  

-22.46783 

14.18750 

-8.63533 

-8.33040 

3.17025 

2  .  86532 

2485  

-22.07325 

14.41140 

—8.48465 

-8.20021 

3.01957 

2.73513 

3719  

—23.79901 

14.58650 

—8.38551 

—8.12100 

2  .  92043 

2  .  65592 

4564  

-23.60079 

14  .  67542 

-8.27621 

-8.09364 

2.81113 

2  .  62856 

5526  

-23.46848 

14.75848 

-8.22696 

-8.05470 

2.76188 

2.58962 

6604  

-23.25572 

14.83588 

-8.09160 

-8.04626 

2  .  63652 

2.58118 

7629  

-23.13332 

14  .  89854 

-8.03186 

-8.01561 

2.56678 

2.55053 

Below  the  Isothermal  Layer. 

7687..  

-23.12281 

14.90183 

-11.88806 

-11.80980 

0.42298 

0  .  34472 

8369  

-23.01921 

14.93874 

-11.82137 

-11.78979 

0.35629 

0.31471 

9443  

-24.94727 

14.99118 

-11.80187 

-11.77086 

0.33679 

0.30578 

10511  

-24.85529 

15.03771 

—  11.75662 

—11.75561 

0.29154 

0.29053 

11552  

-24.78105 

15.07848 

-11.72295 

-11.73234 

0.25687 

0.26726 

12572  

-24.69859 

15.11548 

-11.67749 

-11.72891 

0.21241 

0.26383 

13405 

-24  .  63058 

15.14335 

-11.63735 

-11.71841 

0.17227 

0.25333 

The  entire  system  is  reduced  to  the  equivalent  of  the  ex- 
ponent 4.00,  as  in  the  Stefan  Law. 


THE   WIEN   DISPLACEMENT   LAW  319 

In  the  terrestrial  section  I  the  divisor  A  c  .  y3  consists  of  the 
mean  factor  Ac  =  2.13658  required  to  reduce  from  the  coeffi- 
cients corresponding  with  the  exponent  2.42  to  that  for  4.00 
and  73.  Since  A  c.y3  is  equal  to  a  little  more  than  y4  =  y4/3  x  3  for 
hz  in  the  equation,  it  is  possible  that  y4  is  the  fundamental  reduc- 
tion for  h  from  the  solar  adiabatic  layer  to  the  terrestrial  non- 


the  Bigelow  (~TJ\   ,  and  the  Planck  \~rf\  ,  as   derived   from 


adiabatic  layer.     The  values  of  7-=  are  thus  nearly  the  same  in 

,  and  the  Planck  \~rf 

their  respective  formulas  for  each.     The  Planck  formula  is  some- 
what indeterminate,  because  it  contains  the  unknown  coefficient 

(48  TT  a    k*\  1/3 
---  i)    y   vm  refers  to  the  frequencies 
(J,  Cr    ' 

2891 
for  the  maximum  wave  lengths,  Xm  =          . 


The  Variations  in  the  Wien  Displacement  Law 
The  Wien  Displacement  Law  is  reduced  to  the  form  \m  T  = 
—  .  -j-j  and  since  (h,  k)  are  variables  it  follows  that  \m  T  must 

p          K 

vary  with  them.  For  the  selected  temperatures  T,  compute 
-r  and  \m  T  in  succession,  and  finally  the  corresponding  maximum 

fv 

wave  length  ^m  in  terms  of  (/A).  On  multiplying  \m  T,  in  \(cm), 
the  displacement  constant  is  about  0.1465  in  the  sun's  non- 
adiabatic  strata;  about  0.1185  in  the  sun's  adiabatic  strata, 
about  0.2891  in  the  higher  levels  of  the  earth's  atmosphere,  but 
gradually  diminishing  to  about  0.1500  near  the  sea  level.  This 
suggests  that  the  Wien  Displacement  Law  for  (h .  k)  variables 
is  only  relatively  a  constant  under  special  thermodynamic  con- 
ditions, and  that  these  differ  from  one  body  to  another.  The 
law  was  deduced  as  a  function  of  (X.T),  but  it  must  be  made  a 
more  complex  function  of  (h .  k  .  X  .  T).  This  difficult  subject 
will  require  further  investigations. 


320  A   TREATISE   ON   THE    SUN'S    RADIATION 

TABLE  104 
THE  WIEN  DISPLACEMENT  LAW,  \m  T  =  -«•  .-T-. 


r 

h 
k 

Am  T 

A™  (M) 

T 

h 
k 

Am  T 

Am  (M) 

Sun's  Atmosphere 

Earth's  Atmosphere 

220.    . 

-11.40081 

3.18137 

6.912 

52.0 

-10.44006 

4.22125 

320.00 

283.    . 

-11.39881 

3.18000 

5.348 

86.7 

-11.91343 

3.69462 

45.35 

679. 

-11.38978 

3.17097 

2.183 

109.3 

-11.71079 

3.49198 

28.40 

1484. 

-11.39576 

3.17695 

1.013 

216.3 

-11.62442 

3.40561 

11.76 

2485. 

-11.37677 

3.16796 

0.592 

222.5 

-11.55179 

3.33298 

9.68 

3719. 

-11.38766 

3.16885 

0.388 

217.3 

-11.48131 

3.26250 

8.42 

4564. 

-11.39555 

3.17674 

0.329 

257.0 

-11.42169 

3.20288 

6.21 

5526. 

-11.38639 

3.16758 

0.266 

279.8 

-11.40245 

3.18364 

5.46 

6604. 

-11.36788 

3.14907 

0.213 

285.5 

-11.39497 

3.17616 

5.26 

7629. 

-11.34555 

3.12674 

0.176 

7687. 

-11.33502 

3.11621 

0.170 

8369. 

-11.32089 

3.10208 

0.151 

9443. 

-11.30074 

3.08193 

0.128 

10511. 

-11.28443 

3.06562 

0.111 

11552. 

-11.25189 

3.03308 

0.094 

12572. 

-11.25376 

3.03495 

0.086 

13405. 

-11.24639 

3.02758 

0.080 

c  =  3  X  1010  10.47712;    0  =  4.9651   0.69593;    — ,    13.78119   (M) 

Another  Formula  for  the  Quantity  hB,  Computed  from  Thermody- 

namic  Data 

We  will  proceed  to  develop  a  series  of  formulas  which  will 
connect  together  definitely  the  radiation  function  that  has  been 

used,  -         — ,  and  the  thermodynamic  potential  and  kinetic 

energies,  together  with  the  radiation  terms  which  enter  the 
Wien-Planck  formula.  Furthermore,  it  will  be  possible  to 
derive  the  terms  of  the  ionization  which  make  the  free  electricity 
of  atmospheres,  and  its  relation  to  the  variable  potential  energy 
h,  and  the  wave  length  or  the  wave  frequency  related  thereto. 
It  may  be  stated  that  there  is  a  complete  confirmation  of  the 
results  of  the  discussion  of  the  radiation  as  developed  in  the 
preceding  chapters,  so  that  the  data  are  thoroughly  self-con- 
sistent, and  can  be  derived  from  at  least  three  independent 
methods  of  computation.  They  throw  much  light  upon  the 
problems  of  atomic  physics,  which  will  be  briefly  mentioned. 


BIGELOW'S    SECOND    THEORY    OF    RADIATION  321 

The  Planck  formula  for  the  Wirkungsquantum  hp  was  de- 
rived from  a  statistical  analysis  of  the  probable  distribution  of 
the  elementary  oscillators,  coupled  with  the  thermodynamic 
data.  We  have  attempted  to  produce  similar  results  by  re- 
ferring to  the  pressure  distribution  of  the  electrons  in  the  atoms 
of  different  weights.  There  has  evidently  been  a  partial  success, 
since  one  branch  of  the  solar  curve  has  been  obtained.  There 
was,  however,  a  difference  in  the  extension  to  the  adiabatic 
strata  in  the  sun,  and  there  was  a  wide  discrepancy  in  applying 
the  formula  to  the  earth's  atmosphere. 

Another  formula  has  been  derived  from  the  fundamental 
equations  of  thermodynamics,  in  which  it  is  the  molecules 
that  are  considered  as  the  units,  rather  than  the  atoms  or  elec- 
trons. The  purpose  has  been  to  compute  the  amount  of  the 
potential  energy  which  is  expended  in  one  oscillation  of  a  molecule 
during  the  passage  of  one  wave.  Hence,  the  rate  of  the  change 
of  the  potential  energy  per  unit  variation  in  the  volume,  the 
number  of  molecules  in  the  unit  volume,  and  the  wave  frequency 
are  the  significant  terms.  A  large  number  of  primary  formulas 
are  developed  in  terms  of  n,  the  number  of  molecules  per  unit 
volume,  and  Ny  the  nurhber  of  molecules  per  unit  mass.  It  is 
possible  to  develop  the  Boyle-Gay  Lussac  Law  and  the  First 
Law  of  Thermodynamics  in  terms  of  n  and  N,  SQ  that  these  two 
fundamental  quantities  are  the  proper  bases  for  all  thermody- 
namic discussions  of  the  problems  of  radiation. 

The  Kinetic  and  the  Potential  Energies  in  Radiation,  Determined 
from  Thermodynamics 

It  is  evident  that  the  complete  relations  between  the  radia- 
tion and  the  thermodynamics  of  atmospheres  must  be  expressed 
in  the  terms  of  the  kinetic  and  the  potential  energies.  Specifically, 
it  is  necessary  to  compute  the  values  of  k,  h,  in  the  Wien-Planck 
formulas  directly  from  the  primary  atmospheric  quantities. 
The  following  development  shows  how  the  Boyle-Gay  Lussac 
Law  and  the  First  Law  of  Thermodynamics  can  be  expressed 
in  terms  of  n,  the  number  of  molecules  in  the  unit  volume, 
and  N,  the  number  of  molecules  in  the  unit  mass,  in  place  of 


322  A   TREATISE   ON   THE   SUN'S    RADIATION 

the  usual  P,  p,  R,  T.    We  have  the  following  thermodynamic 
definitions  : 

(267)  Boyle-Gay  Lussac  Law.     P  =  pRT. 

(268)  Pressure,    P  =  nkT.    dP  =  k  T.dn,  for  kT=  constant. 

n  (dn       n      dN\ 

(269)  Density,      p  =  m  -j.     d  p  =  m  (-^r  -  -^  .  —)  . 

1    N  1   fdN       N  dn\ 

(270)  Volume,      v  =  --  .  dv  =  —  (  -----  ). 

m    n  m  \  n         n     n  ' 

(271)  Efficiency,  R  =  k  •—-.   dR  =  ~  (kdN  +  Ndk). 

(272)  Kinetic  Energy,  PV=mRT  =  N  .  k  T  =  K  T. 

(273)  Specific  Kinetic  Energy,  Pv  =  RT  =  N  .  ^  =  ^  T. 

Differentiate  and  substitute  in  succession  : 

(274)  Pdv  +  vdP  =  RdT  +  TdR  =  R  d  Ta. 


m    n  m    n     n          m 


m         n 

(277)  RdT=^-NdT=^N^f. 

Ill  ill  J. 

(278)  TdR=  —  (kdN  +  Ndk)  =  —  (dN  +  N^ 

m  v  m   V  k 

/«-    \  kT    (  J  *rdn     ,      *rdn\  kT  j 

(279)  Pdv+vdP  = [d  N  —  N h  ^  —  )  =  —  d  N. 

m    V  n  n  /        m 

^      '  m    \      T  k  '        m 

The  last  step  comes  from  k  T  =  Constant,  so  that, 


This  checks  the  differentiated  equation. 

Integrate  each  term  for  the  First  Law  of  Thermodynamics: 

(282)  Work,   (Wi  -  Wo)  =  Pw  (f,  -  n,)  =  (      -  No)  - 


-l 
J 


BIGELOW'S    SECOND   THEORY   OF    RADIATION  323 

(283)  Hydrostatic)  P^Po  =  =  k_T         «,-*, 

pressure,    J       p10  m  «w 

(284)  Efficiency     I,  R1Q  (T,  -  T0)  =  —  N10  Ti  ~  TQ. 

m  IIQ 

(285)  Efficiency   II,  Tw  (&  -  R0)  =  —  F(#,  -  #0)  +  #10 

m   i— 


(286)  Efficiency  III,  i?io  (Z\  -  T0)  +  Ti0  (A  -  #>)  =  ^10 

.      (Ta-TQ)=^(N,-NQ\ 

(287)  Free  Heat,  Ql  -  Q0  =  Cpa  (Ta  -  T0)  -  Cp1Q  (Ta  -  T0). 


-To 


ao  K  —  1 

»io  k  T    Ni-  No 
nio 


=   -  g  fa  -  20)  +  K  nw  k  T  (vi  -  V0). 

It  is  easily  proved  that, 

dn          K      dN  I        I    dN 


/oon\         l~O  i  —       o 

(289)  •     -  =  -   —  -  —  5^  -  .»!  —  %=  . 

nw         K  —  1      Nio  K  —  1    m 

which  were  used  in  the  last  transformations. 

(290)  Work,  Wi-WQ  =  Rlo  (Ta  -  T0)  -  Cp10  (Ta  -  T0). 


nwkT 


K  —  1  m  HIQ 

=  HIQ  k  T  (Vi  —  Z>0). 

Total  Inner  energy 
(291)  Ui-U,  =  Cpa  (Ta  -  To)  -  ft,  (r.  -  To). 

=  -  g  (Zl  -  z»}  -  ~  (N,  -  No). 


.  Ni  -  No 

-r  (*-*)-  -—  -;£-. 

=  —  g(zi  —  z0)  +  HIQ  k  T  (K  —  1)  fa 


324  A   TREATISE    ON    THE    SUN*S    RADIATION 

From  the  last  equation  the  radiation  function  is  found: 
Radiation  function 


rr  f  \  ™  i°  f  i\  L  T> 

=  AW  =  g  («i  -  Zo)  K  .  m  -TT-   .  -  -  +  (K  -  1)  nw  k  T 

-tVio       HI  —  HQ 

=  g  Pio  (21  —  Zo)  K  -          -  +  (K  —  1)  ttio  &  r. 

Wi         WQ 

The  density  of  the  radiation  is  the  potential  energy  together 
with  the  kinetic  energy  for  the  mass  in  the  column  (zi  —  z0), 
g  PIO  (21  —  Zo)  =  Mioj  which  contains  HIQ  molecules.  We  may 
assume  that  this  potential  energy  is  also  expressed  by  n10  h  v, 

where  v  =  the  frequency,  v  =  •—-,  and  h  is  the  potential  energy 

A 

per  oscillation  in  the  path  of  the  ray.     Hence,  we  have,  using  the 
first  form, 

(293)  n10  h  v  =  +  g  ,     ",,  so  that, 

(fli  —  VQ) 

(294)  ^  =  +,fe^).A.l. 

5  (Vi  -  VQ)     nw      v 

If  we  assume  the  Wien  displacement  law, 

(295)  vm  Tm  =  0.2891  (Planck),  we  have, 

rr\ 

(296)  ""•  =  i  =  '     HenC6' 

(297)  ,=  +g 

6 


where  the  temperature  is  that  of  Tm  = 

Va 


c    m 
0  2891 


(AQ-r^xa       J?    a 
^  -)     •  -^  ...........  ......  Planck's  formula. 

(299)  hB  =  g  /2l  ~  ^    •  —  •—..  .  .Bigelow's  formula. 

6  (fi  -  Po)w      n      vm 

hB  is  the  change  in  the  potential  energy  per  unit  volume,  per 
molecule,  per  wave  vibration,  and  it  therefore  is  the  mean  am- 
plitude of  the  potential  energy  of  each  molecule  in  one  oscillation. 


BIGELOW'S    SECOND   THEORY   OF    RADIATION 


325 


Planck's  Wirkungsquantum  should  be  interpreted  in  this 
manner.  The  values  of  h  increase  with  decreasing  temperatures, 
and  h  is  a  wide  variable.  vm  can  be  computed  for  each  Tm. 

The  diagram  of  hB  and  hp,  Fig.  32,  shows  that  the  curves  are 
similar  except  that  the  amplitude  of  hp  is  considerably  greater 
than  that  of  hB  at  the  isothermal  layer. 

TABLE  105 
THE  MOLECULAR  POTENTIAL  ENERGY  IN  THE  EARTH'S  ATMOSPHERE 


Balloon  Ascension,  Uccle,  September  13,  1911. 


C.  G.  S. 


g  (zi  -  «o) 


r 

.  —  .  —   for 

n     vm 


C.Tm 

0.2891 


ATMOSPHERIC  AIR,  m  =  28.736 

ATMOSPHERIC  AIR,  m  =  28.736 

r 

LoghB 

Loghp 

T 

LogAs 

Log  hp 

0 

6... 
11  
16  
21  
25  
29  
33  
37  
41  
45  
49... 
53  
57  
61  

65... 
68 

-23.26522 
-24.54772 
-24.36712 
-24.18255 
-24.02646 
-25.88035 
-25.75124 
-25.63559 
-25.53110 
-25.43576 
-25.34825 
-25.26741 
-25.19222 
-25.12184 

-25.10459 
-26.99386 
-26.93365 
-26.87620 
-26.82071 
-26.76907 
-26.71771 
-26.66646 
-26.61849 
-26.57217 
-26.52723 
-26.48460 
-26.44156 
-26.40049 
-26.36057 

-26.32076 
-26.28411 
-26.24687 
-26.20997 
-26.17416 
-26.14021 
-26.10516 
-26.07145 
-26.04097 
-26.01089 
-27.98406 
-27.95397 
-27.93361 
-27.91444 

-24.87100 
-24.38936 
-24.06853 
-25.83527 
-25.67431 
-25.53936 
-25.42188 
-25.31896 
-25.22653 
-26.14329 
-25.06905 
-25.00024 
-26.93678 
-26.87784 

-26.82108 
-26.77527 
-26.73127 
-26.68964 
-26.64962 
-26.61150 
-26.57385 
-26.53964 
-26.50511 
-26.47190 
-26.44065 
-26.40964 
-26.38119 
-26.35157 
-26.32412 

-26.29646 
-26.27335 
-26.25144 
-26.22934 
-26.20745 
-26.18839 
-26.16340 
-26.13304 
-26.09912 
-26.05750 
-26.00915 
-27.94784 
-27.90550 
-27.87792 

0 

215  0    . 

-27.89619 
-27.87693 
-27.85846 
-27.83852 
-27.81995 
-27.80147 
-27.78081 
-27.75996 
-27.74117 
-27.72689 
-27.70765 
-27.68779 
-27.66777 
-27.64765 

-27.62869 
-27.61086 
-27.59003 
-27.57062 
-27.55197 
-27.53594 
-27.51127 
-27.49137 
-27.47081 
-27.45316 
-27.43505 
-27.41406 
-27.39562 
-27.37797 
-27.36947 

-27.34308 
-27.32628 
-27.30934 
-27.29594 
-27.28524 
-27.27666 
-27.26710 
-27.25455 
-27.24332 
-27.19248 
-27.17990 

-27.86071 
-27.85048 
-27.84036 
-27.83007 
-27.82077 
-27.81150 
-27.80257 
-27.79344 
-27.78630 
-27.77903 
-27.77231 
-27.76609 
-27.76096 
-27.75494 

-27.74904 
-27.74384 
-27.73645 
-27.73111 
-27.72509 
-27.72083 
-27.71711 
-27.70810 
-27.70350 
-27.69099 
-27.67917 
-27.66168 
-27.64418 
-27.62738 
-27.60787 

-27.59043 
-27.57564 
-27.56082 
-27.55130 
-27.53482 
-27.52762 
-27.52060 
-27.51321 
-27.50851 
-27.50438 
-27.49951 

216  7 

218.5  

220.0  
221  0 

222.0  
223  0.  . 

223  5 

224.0  
223  6 

223  .  1  .  . 

222.8  
221.7  
221.2  

220  4 

220.1  
219.8  
219.0.  . 
218.0  
217.1.  . 

71  

74  
77 

80  

83  
86 

216  0 

216.5  
217  .  1  
218.3  
220.5  
227.0  
233.7  
240.3  
248.1  

255.1  
261.7.  .  . 
267.4  
270.9  
278.4.  .  . 
281.4  
284.5  
287.2  
289.0  
290.4  
292.5  

89... 

92  
95  

98  
101  
104  
107 

110..  . 
112 

114  

116  
118 

120  
124  
130  
138..  . 

150  
166  
187. 

202 

211  

326 


A    TREATISE   ON    THE    SUN'S    RADIATION 


TABLE  106 
THE  MOLECULAR  POTENTIAL  ENERGY  IN  THE  SUN. 

r  r    c   i,         g  («i  —  go)      11,  c        c  .  Tm 

C.  G.  S.  KB  —  -, r —  .  —  .  —    for    vm  —  —  =  „  rtor,, 

(vi-vo)io      n     vm  \m      0.2891 


HYDROGEN  1.00 

HYDROGEN  2.00 

HELIUM  4.00 

CARBON  12.00 

r 

Log*a 

T 

LoghB 

r 

LogfeB 

r 

Log/»B 

160 

-20  .  10922 

250 

-21.94535 

245 

-21.86380 

240 

-21.99583 

330 

-21.52664 

415 

-21.21621 

400 

-21.90457 

500 

-21.39246 

500 

-22.94440 

490 

-21.29088 

585 

-22.70645 

610 

-21.18744 

675 

-22.49559 

750 

-22.94654 

765 

-22.35991 

740 

-22.86200 

860 

-22.14042 

850 

-22.72569 

955 

-23.99073 

1025 

-22  .  54442 

1055 

-23.85296 

1000 

-22.46554 

1155 

-23.73776 

1120 

-22.25270 

1300 

-22.21134 

1260 

-23.65236 

1270 

-22.14394 

1365 

-23.50624 

1475 

-23.40681 

1430 

-23.89265 

1575 

-23.90338 

1585 

-23.31500 

1550 

-23.87751 

1700 

-23  .  15858 

1850 

-23.63666 

1815 

-23.15270 

1830 

-23.64833 

1790 

-23.62562 

1935 

-23.07209 

2055 

-23.00079 

2150 

-23.51698 

2180 

-24.93299 

2110 

-23.45087 

2190 

-23.38746 

2305 

-24.86936 

2475 

-23.34117 

2435 

-24.80839 

2400 

-23.27727 

2565 

-24.72080 

2630 

-23.19183 

.2 

2695 

-24.69632 

2710 

-23.12253 

2 

2800 

-23.18871 

2835 

-24.64426 

35 

2985 

-24.59380 

3040 

-24.98431 

3150 

-23.05115 

3145 

-24.52125 

3110 

-23.02073 

'is 

3315 

-24.50106 

•§ 

3500 

-24.92993 

3495 

-24.45832 

3390 

-24.86087 

3620 

-24.97693 

3 

3685 

-24.41730 

3780 

-24.73894 

^J 
• 

3900 

-24.81793 

3885 

-24.37848 

g 

4095 

-24.34152 

4200 

-24.64921 

4160 

-24.75180 

& 

4300 

-24.71832 

4315 

-24.30639 

4545 

-24.27298 

4700 

-24.62845 

4785 

-24.24133 

4680 

-24.55788 

4720 

-24.64203 

5045 

-24.21100 

. 

5150 

-24.54440 

5325 

-24.18455 

5180 

-24.47627 

5300 

-24.54450 

5600 

-24.46466 

5610 

-24.15442 

5890 

-24.12830 

5700 

-24.40198 

5900 

-24  .  45604 

6100 

-24.39754 

6160 

-24.10343 

6230 

-24.33423 

6420 

-24.07941 

6450 

-24  .  37770 

6600 

-24.33859 

6660 

-24.04641 

6750 

-24.27258 

6880 

-24.02639 

7000 

-24.31002 

7100 

-24.27664 

7218 

-24.00748 

7300 

-24.21116 

^ 

7450 

-24.21528 

7500 

-25.98655 

7500 

-24.23340 

c 

7650 

-24.16071 

7685 

-25.96720 

7600 

-24.15831 

rt 

7690 

-24.10892 

7680 

-25.95259 

7715 

-24  .  10382 

7715 

-24.16797 

>-* 

7701 

-24.05411 

7675 

-25.93409 

7705 

-24.05359 

7705 

-24.10742 

13 

7691 

-24  .  00129 

7670 

-25.90632 

7695 

-24.00160 

7695 

-24.04491 

£ 

7681 

-25.94928 

7665 

-25.88558 

7685 

-25.94759 

7685 

-25.97958 

fc 

7671 

-25.89508 

7660 

-25.82240 

7675 

-25.89577 

7675 

-25.91712 

-*j 

7661 

-25.84133 

7652 

-25.90050 

7665 

-25.83907 

7665 

-25.84929 

1 

7652 

-25.72315 

7662 

-25.99194 

7652 

-25.78725 

7652 

-25.78587 

7665 

-25.75970 

7676 

-25.78850 

7640 

-25.74790 

7640 

-25.70234 

7652 

-25.71170 

7665 

-25.67275 

7750 

-25.66468 

« 

7690 

-25.73580 

7690 

-25.63693 

7680 

-25.73808 

v 

S 

8000 

-25.59774 

35 

8476 

-25.67732 

8500 

-25.56087 

8500 

-25.68037 

8550 

-25.59687 

8951 

-25.65733 

9187 

-25.52482 

9451 

-25.63315 

9691 

-25.53836 

•£ 

9427 

-25.63427 

9874 

-25.49587 

1 

9902 

-25.62351 

10561 

-25.46506 

10402 

-25.59186 

10833 

-25.49080 

10378 

-25.59313 

11248 

-25.43779 

11353 

-25.55426 

•3 

10853 

-25.57395 

11935 

-25.42004 

12304 

-25.51977 

11974 

-25.44933 

*£ 

13255 

-25.48769 

13115 

-25.40844 

14207 

-25.45774 

14257 

-25.41550 

15158 

-25.42963 

15398 

-25.39486 

BIGELOW'S    SECOND   THEORY   OF    RADIATION 


327 


TABLE  106—  Continued 
THE  MOLECULAR  POTENTIAL  ENERGY  IN  THE  SUN. 


CALCIUM  40.  00 

ZINC  64.85 

CADMIUM  111.51 

MERCURY  198.41 

T 

LoghB 

T 

Log/»B 

r 

Log/»B 

T 

LoghB 

230 

-21.55292 

300 

-21.40258 

300 

-21.51639 

420 

-21.14156 

550 

-22.72159 

600 

-22.76097 

600 

-22.86002 

640 

-22.68030 

900 

-22.37932 

890 

-22.30522 

900 

-22.44028 

850 

-22.35401 

1200 

-22.03800 

1180 

-22.00821 

1200 

-22  .  06172 

1150 

-22.01314 

1500 

-22.75701 

1480 

-23  .  77339 

1500 

-23.74720 

1450 

-23.72689 

1850 

-23.51871 

1790 

-23.57788 

1800 

-23.48390 

1800 

-23.48438 

2200 

-23.32328 

2110 

-23.41177 

2150 

-23.25608 

2150 

-23.28636 

2440 

-23.26827 

2550 

-23.15603 

2500 

-23.07288 

2550 

-23.11613 

2780 

-23.14226 

2950 

-23.00686 

2900 

-24.97003 

3000 

-24.97029 

3120 

-23.03067 

3350 

-24.87851 

5 

3460 

-24.93021 

3400 

-24  .  76029 

3450 

-24  .  84759 

B 

3750 

-24.76517 

C 

C/3 

3800 

-24  .  83917 

3900 

-24.68863 

3950 

-24.73723 

u 

4200 

-24.66204 

4140 

-24.75554 

'£ 

4480 

-24.67869 

JS 

4400 

-24.52947 

4450 

-24.64143 

S 

4700 

-24.56977 

4830 

-24.60720 

1 

5000 

-24.43197 

4950 

-24.55546 

5225 

-24.48668 

5190 

-24.54078 

§ 

5560 

-24.47676 

5450 

-24.48681 

z 

5750 

-24.39883 

5940 

-24.42049 

5700 

-24.34787 

5900 

-24.40644 

6300 

-24.33804 

6320 

-24.36625 

6350 

-24.33924 

6700 

-24.31533 

6600 

-24.26731 

6800 

-24.29567 

6750 

-24.27724 

7200 

-24.23752 

7100 

-24.26657 

7300 

-24.20161 

7100 

-24.21855 

u 

7500 

-24.14207 

7500 

-24  .  22694 

7400 

-24.16189 

>l 

7715 

-24.12398 

7745 

-24.17468 

7650 

-24.14205 

7600 

-24.11135 

3 

7705 

-24  .  07731 

7735 

-24.13613 

7670 

-24.06122 

7695 

-24.02532 

7725 

-24  .  09650 

7690 

-24.08396 

7667 

-24.00043 

13 

7685 

-25.96788 

7715 

-24.04370 

7684 

-24.02540 

7663 

-25.94067 

& 

7675 

-25.90878 

7705 

-24.00437 

7677 

-25.96701 

1 

7665 

-25.81193 

7695 

-25.96132 

7669 

-25.90505 

7659 

-25.88993 

7652 

-25.94886 

7685 

-25.91737 

7660 

-25.84675 

7656 

-25.83591 

S 

7675 

-25.87538 

7652 

-25.79100 

7652 

-25.78992 

1—1 

7665 

-25.82680 

7652 

-25.78635 

7640 

-25.74322 

7700 

-25.69642 

7700 

-25.70688 

7720 

-25.73446 

7800 

-25.74041 

d 

8400 

-25.65650 

8500 

-25.67005 

8550 

-25.68319 

2 

9171 

-25.61771 

S 

9600 

-25.57183 

9942 

-25.58258 

9561 

-25.61561 

9494 

-25.63361 

.0 

10713 

-25.55075 

10621 

-25.57053 

10437 

-26.59501 

11504 

-25.49726 

11484 

-25.52071 

11682 

-25.52968 

11381 

-25.55770 

1 

12255 

-25.49272 

12742 

-25.49247 

12324 

-25.52798 

3 

13406 

-25.43299 

13026 

-25.46650 

13803 

-25.45793 

13268 

-25.49152 

•2 

15308 

-25.37649 

14864 

-25.42503 

14211 

-25.46186 

^ 

17210 

-25.32660 

15924 

-25.40965 

15155 

-25.43731 

19112 

-25.28173 

21014 

-25.19050 

328  A   TREATISE   ON   THE   SUN'S   RADIATION 

It  should  be  remembered  that  in  computing, 

«»» 

the  value  of  a  in  the  denominator  is  the  same  as  the  coefficient 
c  in  the  quasi-Stefan  Law 

(301)  KM  =  cTb 

and  that  the  coefficient  c  and  the  exponent  b  were  obtained  by  the 
method  of  trials.  Since  these  vary  together,  and  b  is  about 
4 . 00  in  the  non-adiabatic  layers,  and  about  2 . 42  in  the  adiabatic 
layers,  it  follows  that  the  mutual  adjustment  of  c,  b,  will  produce 
the  same  values  for  h  by  the  Planck  and  the  Bigelow  formulas, 

for  the  same  value  of  -       — .It  will  be  necessary  to  study  more 

vi  -  v0 

fully  the  relations  between  them  before  reaching  any  opinion  as 
to  their  meaning.  But  it  is  clear  that  they  have  the  same  hor- 
izontal branch  in  the  isothermal  layer,  and  that  they  can  be  re- 
duced to  the  same  values  in  the  adiabatic  and  non-adiabatic  layers. 

It  is  seen,  furthermore,  that  if  the  line  radiation  in  the 
spectrum  of  any  element  is  due  to  a  definite  value  of  h,  which 
represents  a  given  pressure  and  temperature,  then  the  series  of 
spectral  lines  will  be  spaced  as  commonly  observed,  by  giving  h 
certain  fixed  increments  in  steps.  In  this  case  the  series  of  lines 
of  an  element  is  due  to  its  independent  radiations  at  such  levels 
as  produce  the  given  P.  T.,  and  if  the  scale  can  be  fixed  these 
lines  can  be  located  in  the  levels  26,  Zi,  z 

In  the  earth's  atmosphere  the  agreement  between  hp  and 
hB  is  very  close,  and  these  small  variations  can  be  adjusted 
through  the  c  coefficient  and  b — exponent. 

Summary 
It  should  be  noted  that  the  hB  equation  in  its  expanded  form, 

/orkoN      7  K  kT  TO.  —  TQ         ,  .  HIQ 

(302)  hB  = 7  .  —  Na  —f .  (K  -  1)  m  n TT  . 

K  —  l      m  l  aQ  iv  i  —  iV  o 


J_    0.2891 

*      x-  T*        ' 
™10          C  1.  rn 

T     TV.  A7  H    OQQ1 

(303)   hB  =  K.kT 


Ta-T0         Na          0.2891 


•  AY-  .#• '   ^  rm  ' 


DERIVATION    OF   CERTAIN   FORMULAS  329 

reduces  finally  to  terms  in  (T,  N),  and  includes  the  Wien  Displace- 
ment Law.  A  new  system  of  physics  can  be  founded  upon 
(T,  n,  N)  since  these  include  the  potential  energy  h,  and  the 
kinetic  energy  k  for  each  oscillation.  Whether  the  molecule  and 
the  atom  are  constructed  as  electric  oscillators  (Planck),  or  as  a 
large  volume  nucleus  of  positive  charges  (Thomson),  or  as  a 
small  condensed  positive  nucleus  (Rutherford),  the  h,  k,  are  not 
coefficients  of  the  internal  structure,  but  of  the  outer  forces  of  col- 
lision between  independent  molecules.  The  secondary  effects  of 
such  collisions  in  producing  electromagnetic  radiation,  and  vibra- 
tory adjustments  of  structures  between  the  positive  and  the 
negative  charges,  lead  to  the  fundamental  problems  in  physics, 
chemistry,  and  the  constitution  of  matter.  It  is  hoped  that 
some  further  progress  can  be  made  in  this  direction  by  the  study 
of  the  spectral  series,  the  atomic  numbers  and  allied  problems, 
from  the  point  of  view  that  h,  k,  n,  N,  are  all  primary  variable 
quantities  in  the  standard  equations  of  thermodynamics  and 
electromagnetics. 

Derivation  of  Certain  Formulas  for  Radiation,  lonization,  and 
Atmospheric  Electricity 

It  seems   that  the  fundamental  formula  for  the  radiation, 

(304)    4r 


is  competent  to  produce  the  Wien-Planck  formula  for  radiation, 
though  the  terms  are  differently  interpreted;  the  fundamental 
relation  between  ionization,  potential  energy,  and  frequency  of 
the  vibration;  and  the  quantity  of  the  electric  charge  in  the 
unit  volume.  Generally,  we  have  for  the  single  #-atom,  of 
which  there  are  nw  in  the  unit  volume,  the  potential  energy  = 

hv  =  g(zi  —  ZQ)   (K  —  1)  m  -^—  '~^r,  and  the  kinetic  energy  = 

IV  1  —  l\o 

(K  —  1)  k  T,  so  that,  for  single  F-atoms, 

(305)  Ul~  U°  =  h  v  +  (K  -  1)  k  T, 

Vi  —  VQ 

as  the  primitive  form,  whereas  Planck  assumed, 

(306)  u  =  Ul~U"  =  hv  +  kT. 


330  A   TREATISE    ON   THE    SUN'S    RADIATION 

This  will  lead  to  the  Planck  form  of  the  function  with  the 
change  of  1  to  K  —  1,  that  is,  0.666  for  the  monatomic  gases 
and  approximately  0.410  for  the  diatomic  gases.  Thus, 


—  (K  —  1) 

In  this  formula  it  is  k  T  which  is  the  variable  constant  in  any 
given  atmosphere,  while  h  v  is  the  variable  potential  energy. 
It  has  become  evident  that  this  potential  energy  is  external 
to  the  atoms  and  molecules,  such  as  occurs  in  the  thermal  colli- 
sions between  atoms  or  molecules  consisting  of  m  unit  charges 
e.  Hence  the  theory  of  the  generation  of  radiation  is  transferred 
from  the  saltum  changes  within  the  circulating  orbits  of  an  atom 
into  the  mutual  repulsions  of  the  atoms  and  molecules  in  the  col- 
lisions which  integrate  into  the  existing  pressure  under  the  force 
of  gravitation.  Evidently  the  potential  energy  changes  from 
level  to  level. 

lonization,  Potential  Energy,  and  Frequency 

It  will  be  convenient  to  make  the  following  transformations: 

1     N 
From  the  equation  for  the  specific  volume,  v  =  --  ,  we 

m    n  ' 

have, 

,      N  1    fdN       N  dn\ 

(308)  d  v  =  —  I  ----  )  .     Integrate, 

m   \  n  n     n  / 

/onnx      /  x  1         Ni  —   N 

(309)  (Vi  —  fl0)  =  — 

m 

=  J_ 
m 


m 

I          1 


ATio 

ni  —  n0\ 

nw 

»10        /' 

nw 

K  -    1            NW 

tf  i  -  No\ 

K  i 

J  j 

No 

K  —  1  '  m          nw 
Substitute  in  the  general  equation  of  radiation  (304) , 

(310)  — =  —  g  -7-^ r-  +  nw  k  T,  for  nw  charges  of  e. 

fli  —  ^o  \y\  —  Vo)w 

Hence   we  shall  assume  that  the  potential  energy  of  one  H- 
atom  in  thermal  collisions  is, 


IONIZATION..  POTENTIAL   ENERGY.    AND    FREQUENCY    .     331 

(311)         hv=-g(Zl-zQ).~.±or, 


(312)         h  v  =  +  g  (zi  -  z0)  (K  -  1)  m 


that  is,  in  terms  of  the  number  of  ^-charges  per  unit  volume  n, 
or  per  unit  mass  N. 

We  can  separate  this  equation  into  the  terms  for  the  ioni- 
zation,  the  potential  energy,  and  the  frequency  changes,  as 
follows : 

Take  the  second  differences  of  (Ni  —  No), 

fhdv  4-  v dh\ 

(313)  d  (Ni  —  No)  =  —  g  (zi  —  Z0)   (K  —  1)  m  ( ^—9 ) . 

v         n  v         ' 

Integrate, 

(314)  A  (Ni  -  No)  =  -  g  (zi  -  z0)  (*  —  1)  f»  JT~ 

Since  for  one  //-charged  atom, 

(315)  (Ni  —  No)  =  +  g  (zi  —  Zo)  (K  —  1)  m  7—-  -  on  the  average, 

A  (Ni  -No)       hi-  ho       vi  -  PQ 


N^No  h1Q  vlQ 

This  equation  is  valid  in  all  atmospheres.     Compare  the  Tables 
107,  108,  and  109. 

Atmospheric  Electricity 

The  potential  equation  is  immediately  applicable  to  the 
problem  of  the  origin  of  atmospheric  electricity  and  the  quantity 
of  electric  charges  in  the  unit  volume  by  adopting  the  Einstein 
formula, 

(317)  h  v  =  Ve. 

The  fundamental  electrostatic  equations  are, 

(318)  Potential  =  V  =  — . 

(319)  Electric  force    =  E  =  -  ^  =  ~. 

d  r        r 

(320)  Work  of  potential  =  W  =  Ve  =—  =  hv. 


332  A   TREATISE    ON   THE    SUN?S    RADIATION 

The  preceding  development  assumes  that  the  fundamental 
charges  (e  .  e)  are  alike,  bemg  in  repulsion  during  the  collisions 
and  along  the  free  path,  so  that  we  have  only  to  transform  the 
expressions  for  h  v  into  volts  per  unit  distance  in  order  to  com- 
pute the  accumulated  charge  in  the  stratum  (z2  —  Zi). 

W 

(321)  Q  =  -^-  =  (F2  -  FO  (*,  -  Zl). 

We  have  1  E.  S.  U.  =  300  volts. 

1  Volt   =  0.00333  E.  S.  U. 


=  0.00333  .    Hence, 

cm.  3 

,      ,  Volts          Volts 

(322)  -  =  —  -- 

meter       100  cm. 

Preliminary  trials  on  the  data  of  atmospheric  electricity 
indicate  that  m  must  be  identified  with  the  molecular  weight 
in  the  earth's  atmosphere,  m  =  28.74,  approximately.  Thence 
the  formula  for  Q  in  successive  strata  gives  the  quantities  which 
when  summed  together  produce  the  electric  charges  on  each 
level  above  the  plane  of  reference,  as  the  earth's  surface  or  the 
top  of  the  sun's  adiabatic  strata.  The  results  are  indicated  in 
Tables  110  and  111. 

Tables  107,  108,  and  109  give  the  results  for  the  terms, 

—  T  --  ,  the  rate  of  change  of  the  potential  in  the  stratum, 

"10 

—  ,  the  rate  of  change  of  the  frequency  in  the  stratum, 

—  A  ^  _        ,  the  rate  of  change  of  the  ionization  in  the 
stratum, 

for  the  balloon  ascension,  Uccle,  September  13,  1911,  in  the 
terrestrial  atmosphere,  and  for  helium  and  zinc  in  the  sun's 
atmosphere,  the  other  elements  being  similar  in  their  develop- 
ment. Figs.  33  and  34  show  the  relative  values  for  the  earth's 
atmosphere,  m  =  28,  and  for  the  solar  atmosphere  of  helium, 


ATMOSPHERIC   ELECTRICITY  333 

m  =  4.00.  The  potential  is  the  value  which  occurs  in  each 
wave  length  and  its  oscillation  so  that  it  can  be  analyzed  accord- 
ing to  the  formulas  of  Chapter  I.  The  ionization  term  marks 
the  disintegration  of  the  mass  according  to  the  changes  in 
the  density  per  unit  mass.  These  data,  therefore,  become  of 
primary  importance  in  studies  of  the  atomic  physics. 

In  the  earth's  atmosphere  the  convectional,  isothermal,  and 
non-adiabatic  regions  are  well  defined.  At  the  top  of  the 
isothermal  strata  there  is  a  rapid  change  in  the  frequency  and 
ionization,  while  the  potential  increases  quite  steadily  in  value. 

In  the  solar  atmosphere  there  is  no  ionization  in  the  adia- 
batic  strata;  there  are  sudden  large  changes  in  the  frequency 
and  ionization  at  the  isothermal  layer,  including  the  photosphere, 
where  the  radiation  is  especially  generated;  in  the  non-adiabatic 
region  they  increase  together  to  very  large  values.  Figs.  33,  34. 

It  should  be  remarked  that  Abbot's  value  of  the  solar  constant 
of  radiation,  corresponding  with  the  temperature  5810°,  finds  no 
support  in  these  results.  They  are  in  harmony  with  the  view  that 
the  solar  radiation  originates  at  the  temperature  of  7655°. 

The  Terrestrial  Atmospheric  Electricity 

There  are  many  hypotheses  in  the  literature  of  atmospheric 
electricity  regarding  the  origin  and  the  distribution  of  the  power- 
ful electric  charges  which  are  a  characteristic  and  permanent 
constituent  of  all  atmospheres,  but  they  are  generally  deficient 
and  unsatisfactory.  We  shall  employ  the  Einstein  relation, 
Ve  =  h  v,  as  the  electrostatic  form  of  the  prevailing  potential 
energy  of  radiation,  reducing  it  to  the  form, 
Volts  300 


= h  v.     100  m,  for  voltage,  and 


meter         e 

(Qi  ~  Co)  =  (21  -  z0)  (Fi  -  Fo),  for  quantity. 

Table  110  contains  the  results  for  (Fi-  F0),  (Qi  -  Co)  arranged 
by  the  arguments  z  and  T.  The  values  of  (Fi  —  Fo)  have  three 
distinct  stages: 

(Fi  —  Fo)  varies  between  1.00  and  2.00  in  the  convection 
strata;  it  increases  in  the  isothermal  region  to  about  10.00  at 


334 


A   TREATISE    ON    THE    SUN  S    RADIATION 


TABLE   107 

UCCLE,  SEPTEMBER  13,  1911 
-A  (Ni  -  N0)       hi  -h0 


i  —  No 


z 

hi  -ho 

V\  —  VQ 

Cum 

-A(ATi-ATo) 

Meters 

hio 

V10 

oil  in 

Nt-No 

80000  

o°.o 









78000  

11.0 

0.40997 

-.37017 

.02980 

.34468 

76000..... 

21.0 

.35561 

-  .  17392 

.  18169 

.25819 

74000  

29.0 

.30862 

-  .  12903 

.  17959 

.  18751 

72000  

37.0 

.23946 

-  .  10255 

.  13691 

.  15143 

70000  

45.0 

.20080 

-.08517 

.11563 

.  12360 

68000  

53.0 

.17266 

-.07271 

.09995 

.  10518 

66000  

61.0 

.15453 

-.06348 

.09105 

.10040 

64000  

68.0 

.  14302 

-.04318 

.09984 

.09899 

62000  

74.0 

.12761 

-.03972 

.08789 

.09122 

60000  

80.0 

.11813 

-.03682 

.08191 

.08354 

58000  

86.0 

.11035 

-.03428 

.07607 

.07876 

56000..... 

92.0 

.  10339 

-.03210 

.07129 

.07437 

54000  

98.0 

.09032 

-.03012 

.06020 

.06923 

52000  

104.0 

.09166 

-.02841 

.06325 

.06514 

50000  

110.0 

.08437 

-.01806 

.06631 

.07018 

48000  

114.0 

.08491 

-.01735 

.00756 

.06761 

46000  

118.0 

.07813 

-.01701 

.06712 

.05769 

44000  

124.0 

.07763 

-.04727 

.03036 

.02031 

42000  

138.0 

.06920 

-.08318 

-.01398 

-.02941 

40000 

166.0 

.06925 

-.11897 

-.04972 

-.04545 

38000  

202.0 

.04413 

-.04359 

.00054 

.00177 

36000  

215.0 

.04433 

-.01017 

.03416 

.03823 

34000  

218.5 

.04590 

-.00681 

.03909 

.03830 

32000 

221.0 

.04254 

-.00452 

.03802 

.04068 

30000  

223.0 

.04800 

-.00458 

.04392 

.03799 

28000  

224.0 

.03289 

.00181 

.03470 

.03050 

26000  

223.1 

.04572 

.00149 

.04721 

.04747 

24000 

221.7 

.04633 

.00296 

.04929 

.03775 

22000  

220.4 

.04103 

.00140 

.04243 

.03978 

20000  

219.8 

.04467 

.00360 

.04827 

.04097 

18000  

218.0 

.04230 

.00416 

.04646 

.05027 

16000  

216.0 

.04586 

-.00192 

.04394 

.03674 

14000  

217.1 

.04065 

-.00553 

.03502 

.04092 

12000  

220.5 

.04829 

-.02907 

.01922 

.01405 

10000  

233.7 

.04062 

-.02781 

.01281 

.01156 

8000  

248.1 

.03772 

-.02788 

.00984 

.01234 

6000  

261.7 

.03900 

-.02156 

.01744 

.01901 

4000 

270.9 

.02463 

-.02730 

-.00267 

-.00497 

2000  

284.5 

.02869 

.00937 

.01932 

.01986 

1000  

289.0 

.02690 

-.00486 

.02204 

.01975 

500  

290.4 

— 

— 

— 

i— 

100  

292.5 

— 

— 

— 

— 

IONIZATION,    POTENTIAL   ENERGY   AND   FREQUENCY          335 


TABLE  108 
HELIUM,  m  =  4.00 

-A  (ft  -  No)       hi-h* 

Ni-  No  &io 


z 

hi  -ho 

VI  —  VQ 

Cum 

-A(Ni-No) 

Kilometers 

km 

via 

ollltl 

Ni-No 

11000  

240° 

1.34090 

-.94892 

0.39198 

1.46400 

10500  

490 

.91438 

-.40648 

.50790 

2.75790 

10000 

740 

.85434 

-.29885 

.  55549 

.  91804 

9500  

1000 

.70840 

-.23779 

.47061 

.66130 

9000  

1270 

.59495 

-  .  19963 

.39532 

.57256 

8500  

1550 

.51579 

-  .  16564 

.35015 

.42196 

8000 

1830 

.  44697 

-.14211 

.30486 

.35483 

7500  

2110 

.39452 

-  .  12863 

.26589 

.29792 

7000  

2400 

.35252 

-  .  12130 

.23122 

.25232 

6500  

2710 

.31562 

-.11484 

.20078 

.21132 

6000  

3040 

.28231 

-.10883 

.17348 

.18633 

5500  

3390 

.27827 

-.10878 

.16949 

.15458 

5000 

3780 

.20587 

-  .  10526 

.10061 

.11610 

4500  

4200 

.20952 

-.10886 

.10066 

.10376 

4000  

4680 

.19040 

-.10141 

.08899 

.08683 

3500 

5180 

.  17294 

—  .09560 

.07734 

.07686 

3000  

5700 

.15337 

-.08885 

.06452 

.06935 

2500 

6230 

.J4159 

-.08009 

.06150 

.06373 

2000  

6750 

.  14122 

-.07850 

'  06272 

.07444 

1500 

7300 

.  12146 

—  .04025 

.08021 

.  10524 

1000 

7600 

.  12532 

—  .01503 

.11028 

.  13702 

500  

7715 

.11500 

.00215 

.11715 

.11000 

0.... 

7705 

.11953 

.00045 

.11998 

.11217 

-500.  .  .  . 

7695 

.  12139 

.00133 

.  12272 

.11282 

-1000.... 

7685 

.11918 

.00129 

.  12047 

.15327 

-1500...  . 

7675 

.  13037 

.00132 

.  13169 

.  12795 

-2000.... 

7665 

.11918 

.00174 

.  12092 

.  13338 

-2500...  . 

7652 

.11311 

-.00371 

.10948 

.06938 

-3000.... 

7680 

.  13272 

-  .  10136 

.03136 

.00805 

-3500.... 

8500 

.10862 

-  .  10595 

.00267 



-4000.... 

9452 

.09498 

-.09579 

-.00081 

— 

-4500...  . 

10402 

.08654 

-.08744 

-.00090 

— 

-5000.... 

11353 

.07939 

-.08041 

-.00102 

— 

-5500.... 

12304 

.07382 

-.07437 

-.00056 

— 

-6000.... 

13255 

.06892 

-.06927 

-.00035 

— 

-6500.... 

14206 

.06473 

-.06478 

-.00005 

— 

-7000.... 

15158 

— 

— 

— 

— 

The  curves  for  the  other  solar  elements  are  similar. 


336 


A   TREATISE    ON    THE    SUN  S    RADIATION 


TABLE  109 

ZINC,  m  =64.85 

-A  (Ni  -  No)  =  hi -ho 
Ni  —  No  h10      "*" 


T 

hi  -ho 

vi  —  vo 

Cum 

-A  (Ni  -  No) 

hio 

VlO 

Ni-No 

650  
625  

230° 
420 

0.88216 
.  97240 

-0.58461 
-  41512 

0.29755 
55728 

0.35806 
1  28730 

600  
575...  . 

640 
890 

.81372 
65837 

-  .32679 
—  28017 

.48693 
37820 

.74920 
49726 

550  
525 

1180 
1480 

.52790 
44272 

-  .22564 
—  18960 

.30236 
25312 

.37027 
29049 

500  
475  
450 

1790 
2110 
2440 

.37788 
.32768 
28813 

-  .16408 
-  .14508 
—  13022 

.21380 
.  18260 
15791 

.23516 
.  19575 
16789 

425  
400  

2780 
3120 

.25552 
23009 

-  .11528 
—  10338 

.  14024 
12671 

.  14861 
13217 

375  

3460 

.  20909 

-  09363 

11546 

11993 

350  
325... 

3800 
4140 

.  19197 
17649 

-  .08566 

—  07887 

.  10631 
09762 

.  10972 
09979 

300  
275  
250  
225  

4480 
4830 
5190 
5560 

.  16424 
.  15272 
.  14706 
12940 

-  .07519 
-  .07369 
-  .06698 
—  06612 

.08905 
.07903 
.08108 
06328 

.09004 
.08076 
.07448 
06818 

200  
175  
150  
125  
100 

5940 
6320 
6700 
7100 
7500 

.12470 
.11710 
.11246 
.09117 
12020 

-  .06199 
-  .05835 
-  .05799 
-  .03176 
—  05517 

.06271 
.05875 
.05447 
.05941 
06503 

.06330 
.05903 
.05350 
.06189 
08657 

75  
50... 

7745 
7735 

.08871 
.09117 

.00130 
00130 

.09001 
09247 

.  10455 
10450 

25  

7725 

.  12147 

.00130 

12277 

12488 

0  
-  25  
-  50  
—  75 

7715 
7705 
7695 
7685 

.08956 
.09907 
.10111 
09661 

.00130 
.00128 
.00133 
00129 

.09086 
.10035 
.  10244 
09790 

.08666 
.  12484 
.08619 
12522 

-100  
-125  
-150  
—  175... 

7675 
7665 
7652 
7640 

.11174 
.09308 
.09922 
.08363 

.00129 
.00176 
.00151 

—  00782 

.11303 
.09484 
.  10073 
07581 

.  10237 
.  10645 
.  10274 
05130 

-200  
-225  
-250  
-275  
-300  
-325...  .  . 
-350..... 
-375  

7700 
8400 
9171 
9942 
10713 
11484 
12255 
13026 

.11588 
.08925 
.08084 
.07324 
.06916 
.06442 
.06036 

-  .08697 
-  .08775 
-  .08067 
-  .07465 
-  .06946 
-  .05893 
-  .06047 

.02891 
.00150 
.00017 
-.00141 
-.00030 
+  .00549 
-.00011 

.00429 

IONIZATION,    POTENTIAL   ENERGY,    FREQUENCY  337 


I 

80,000 

70,000 

, 

-A(Ni-No) 

_j- 

_(KI_TXO) 

~^^ 

->- 

60,000 

hrhn 

^ 

\ 

\ 

\ 

\     s 

\ 

A 

h  10 

50,000 

Xon 

.adiabatic  R 

egion 

\ 

\  \ 

\ 
\ 

\ 
\ 

40,000 

S 

^••^—  . 

-*^^ 

30,000 

$ 

"  (y   "n" 

T\ 

IA(NI-NO) 

~^r 

NrN0 

20,000 

IE 

othermal  R 

egion 

j 
i 

10,000 

V  / 

000 

Convectk 

nal  Region 

*-e 

\ 

\,'\ 

.30000            .25000            .20000              .15000            .10000             .05000            .00000 

Values  of  fhe  Computed  Ratios 
(The  frequuency  is  plotted  with  the  minus  sign) 

FIG.  33.     lonization,  Potential  Energy  and  Frequency  in  the  Terrestrial 

Atmosphere. 


338 


A   TREATISE   ON   THE    SUN'S   RADIATION 


//• 


t-l 


< 


1 


ATMOSPHERIC   ELECTRICITY  339 

36,000  meters,  but  falls  to  —15.00  at  40,000  meters,  then  rises 
steadily  to  very  large  values  as  1000.00  on  the  vanishing  plane. 
The  values  of  the  computed  quantity  by  summation  range  from 
80  volts  on  the  surface  upwards  to  about  200,000  volts  at  the 
top  of  the  isothermal  layer;  this  value  is  nearly  the  same  through 
the  stratum  34,000  —  45,000  meters;  above  that  level  (Qi  —  Q0) 
increases  to  very  large  values.  Comparing  with  the  average 
charges  observed  in  balloon  ascensions,  Drexel,  Nebraska, 
November  and  December,  1915,  the  increases  in  vertical  value 
are  in  close  accord  up  to  4,000  meters. 

This  distribution  is  in  agreement  with  the  known  conditions, 
generally,  with  increasing  values  in  the  convectional  and  iso- 
thermal region,  a  layer  of  steady  values  at  about  200,000  volts, 
and  a  supercharged  non-adiabatic  region  to  the  vanishing  plane. 
The  curve  of  voltage  follows  closely  that  of  the  free  heat  (Qi—  Qo). 
When  the  charge  overpasses  the  insulating  capacity  of  the  gaseous 
medium  there  is  a  discharge  as  in  accumulations  in  the  lower 
strata  during  temporary  thunderstorms,  or  permanently  in  the 
strata  above  the  isothermal  layers.  This  latter  is,  then,  the 
fundamental  cause  of  the  free  electric  charges  which  go  over 
into  the  polar  auroras  and  the  disturbances  of  the  normal 
magnetic  field  as  was  explained  in  the  Meteorological  Treatise, 
Chapter  VI. 

The  cause  of  atmospheric  electricity  appears  to  be  a  ther- 
modynamic-radiation  effect  which  is  due  to  the  deficiencies  of 
the  non-adiabatic  physical  conditions  relative  to  the  adiabatic 
conditions.  Adiabatic  strata  have  no  free  electricity,  but,  on 
diverging  into  the  non-adiabatic  temperature  gradients,  there 
is  set  up  a  process  of  transformation  of  energy  which  is  of  the 
electrostatic  type,  though  it  is  really  an  integrated  electro- 
magnetic form  of  potential  energy. 

The  Solar  Atmospheric  Electricity 

On  Table  111  are  collected  the  values  of  (Fi  —  F0)  computed 
for  helium  and  zinc  by  the  same  formula.  It  is  seen  that  the 
potential  (Fi  —  F0)  vanishes  in  the  adiabatic  strata;  in  the 


340 


A   TREATISE   ON   THE    SUN  S    RADIATION 

TABLE  110 
TERRESTRIAL  ATMOSPHERIC  ELECTRICITY 


300 


meter 


h  v.  100  m. 


z 

T 

(Fi  -  Fo) 

z 

T 

o,-n, 

S  (Qi  -  Qo) 

Observed  at 
Drexel,  Neb. 
Nov.,  Dec.. 

1915 

1915 

79000... 

6°.0 

— 

36000 

215°  .0 

10.66 

222307 

Elevation 
o9o  JMeters 

78000..  . 

11  .0 

293.3 

35000 

216  .7 

11.01 

211647 

77000..  . 

16  .0 

990.9 

34000 

218  .5 

11.32 

200637 

76000..  . 

21  .0 

1012.1 

33000 

220  .0 

10.66 

189317 

75000... 

25  .0 

723.4 

32000 

221  .0 

10.21 

178657 

03  b 

1 

74000... 

29  .0 

638.3 

31000 

222  .0 

11.11 

168447 

73000... 

33  .0 

491.4 

30000 

223  .0 

11.28 

157337 

J2  c 

72000..  . 

37  .0 

386.0 

29000 

223  .5 

8.69 

146057 

•£  03 

71000.  .. 

41  .0 

310.0 

28000 

224  .6 

8.38 

137367 

II 

70000... 

45  .0 

252.6 

27000 

223  .6 

10.64 

128987 

a 

69000... 

49  .0 

209.0 

26000 

223  .1 

9.82 

117347 

03 

^  "** 

68000... 

53  .0 

175.5 

25000 

222  .8 

10.13 

107527 

'B  h 

0 

67000  .  .  . 

57  .0 

149.6 

24000 

221  .7 

9.30 

97397 

(j  C 

66000.  .. 

61  .0 

132.0 

23000 

221  .2 

8.37 

88097 

*"  £ 

65000.  .  . 

65  .0 

124.9 

22000 

220  .4 

7.30 

79727 

§  £ 

*x 

64000... 

68  .0 

114.4 

21000 

220  .1 

8.09 

72427 

I 

63000  .  .  . 

71  .0 

99.2 

20000 

219  .8 

7.57 

64337 

«  c 

62000.  .  . 

74  .0 

87.84 

19000 

219  .0 

7.09 

56767 

rt  S 

61000.  .  . 

77  .0 

74.02 

18000 

218  .0 

5.84 

49677 

.§  c 

60000  .  .  . 

80  .0 

68.89 

17000 

217  .1 

8.40 

43837 

O  £j 

59000.  .  . 

83  .0 

64.30 

16000 

216  .0 

5.65 

35437 

a  rt 

58000..  . 

86  .0 

54.83 

15000 

216  .5 

5.40 

29787 

§ 

ctf 

57000... 

89  .0 

49.09 

14000 

217  .1 

4.18 

24387 

CJ  .  i 

56000..  . 

92  .0 

44.37 

13000 

218  .3 

3.62 

20207 

v 

nn 

55000..  . 

95  .0 

38.87 

12000 

220  .5 

2.15 

16587 

&.s  § 

54000..  . 

98  .0 

37.35 

11000 

227  .0 

1.46 

14437 

:§> 

53000.  .  . 

101  .0 

33.00 

10000 

233  .7 

1.39 

12977 

J  u 

52000  .  .  . 

104  .0 

30.27 

9000 

240  .3 

1.14 

11587 

H  & 

o 

51000..  . 

107  .0 

28.37 

8000 

248  .1 

1.05 

10447 

o  "£> 

50000..  . 

110  .0 

28.72 

7000 

255  .1 

1.37 

9397 

49000.  .  . 

112  .0 

25.55 

6000 

261  .7 

1.80 

8027 

48000.  .  . 

114  .0 

24.72 

5000 

267  .4 

0.82 

6227 

47000... 

116  .0 

22.31 

4000 

270  .9 

0.73 

5407 

6052 

7232 

46000.  .  . 

118  .0 

19.65 

3000 

278  .4 

0.91 

4677 

4451 

5226 

45000..  . 

120  .0 

14.53 

2500 

281  .4 

1.09 

4227 

4149 

4815 

44000.  .  . 

124  .0 

8.84 

2000 

284  .5 

1.69 

3682 

3519 

4219 

43000.  .  . 

130  .0 

2.99 

1500 

287  .2 

2.09 

2837 

2071 

2970 

42000  .  .  . 

138  .0 

-  4.13 

1000 

289  .0 

1.96 

1792 

946 

1422 

41000.  .  . 

150  .0 

-11.64 

500 

290  .4 

1.83 

812 

127 

240 

40000.  .  . 

166  .0 

-15.32 

100 

292  .5 

— 

80 

— 

— 

39000... 

187  .0 

-  9.69 

38000... 

202  .0 

0.18 

37000  .  .  . 

211  .0 

7.45 

ATMOSPHERIC    ELECTRICITY 


341 


TABLE  111 

SOLAR  ATMOSPHERIC  ELECTRICITY 
V          300 


meter 


h  v.  100  m. 


I 

HELIUM,  m 

=  4 

ZINC 

m  =  65 

z 

T 

(Vi  -  Fo) 

2 

T 

(Vi  -  Fo) 

11000  
10500  

240° 
490 

3.703X107 
1.093X107 

650 
625 

230° 
420 

1.014X108 
1.1632  XlO8 

10000  
9500  

740 
1000 

6.428X106 
3.005X106 

600 
575 

640 
890 

5.371X107 
2.  507  XlO7 

9000 

1270 

1  .  565  X  106 

550 

1180 

1.371  XlO7 

8500  
8000 

1550 
1830 

9.254X106 
5  .  697  X  105 

525 
500 

1480 
1790 

8.438X106 
5  .  676  X  106 

7500  
7000 

2110 
2400 

3.689X105 
2  481  XlO5 

^ 

475 
450 

2110 
2440 

3.892X106 
2  824  XlO6 

6500  
6000  

2710 
3040 

1.635X105 
1.229  XlO5 

o 

3 
P 

425 
400 

2780 
3120 

2.  154  XlO6 
1.696  XlO6 

2 

5500  
5000  
4500 

3390 
3780 
4200 

1.014  XlO5 
5.  204  XlO4 
4  746  XlO4 

Q. 
P" 
1* 

375 
350 
325 

3460 
3800 
4140 

1.367  XlO6 
1.125  XlO6 
9  311X105 

P 

c- 

p- 

4000  
3500 

4680 
5180 

3.668X104 
2  922  XlO4 

300 
275 

4480 
4830 

7.730X106 
6  436  XlO5 

I 

n 

3000  
2500  
2000  
1500  
1000  . 

5700 
6230 
6750 
7300 
7600 

2.433X104 
2.  103  XlO4 
2.  016  XlO4 
2.  421  XlO4 
2  985  XlO4 

250 
225 
200 
175 
150 

5190 
5560 
5940 
6320 
6700 

5.  761  XlO5 
4.  331  XlO5 
4.032X105 
3.551X105 
3  082  XlO5 

500  

0  
-  500  

7715 

7705 
7695 

2.  840  XlO4 

2.619X104 
2  .  333  X  104 

? 

125 
100 
75 
50 

7100 
7500 
7745 
7735 

3.  215  XlO5 
3.228TX105 
4.  285  XlO5 
3  956  XlO5 

-1000  
-1500  
-2000  
-2500  
-3000  

-3500 

7685 
7675 
7665 
7652 
7680 

8500 

2.  019  XlO4 
1.945  XlO4 
1.594  XlO4 
1.248X104 

thermal. 

25 

0 
-  25 
-  50 

-  75' 
—  100 

7725 

7715 
7705 
7695 
7685 
7675 

4.717X105 

3.  167  XlO5 
3.  140  XlO5 
2.900X105 
2.  308  XlO5 
2  607  XlO5 

! 

-4000  
—4500  . 

9452 
10402 

- 

•>* 

-125 
—  150 

7665 
7652 

3.161X105 
0  706  XlO5 

P- 

-5000  
-5500  
-6000  
-6500... 

11353 
12304 
13255 
14206 

- 

Adiabatic. 

-175 

-200 
—225 

7640 

7700 
8400 

1.306  XlO5 
0.473X105 

-7000  

15158 

-250 
-275 
-300 
-325 
-350 
-375 

9171 
9942 
10713 
11484 
12255 
13026 

- 

Adiabatic. 

342  A   TREATISE   ON    THE    SUN*S    RADIATION 

isothermal  region  helium  has  a  nearly  constant  value  of  about 
20,000  volts  per  meter,  and  zinc  350,000  volts;  in  the  non- 
adiabatic  strata  they  increase  to  very  large  values.  This  is  in 
harmony  with  the  distribution  of  the  terrestrial  atmospheric 
electric  charges. 

It  is  noted  that  in  the  adiabatic  layers  where  the  umbra  of 
sun  spots  is  developed,  that  is,  in  the  low-level  primary  vortex, 
there  are  no  free  charges.  The  magnetic  effects  noted  must  be 
due  to  the  charged  atoms  and  molecules,  rather  than  to  the 
free  electricity;  in  this  region  the  Stark  electrical  effect  is  lack- 
ing. In  the  penumbra  and  the  upper  secondary  vortex  there  is 
free  electricity,  but  without  rapid  rotations  and  no  Stark  effect. 
The  upper  levels  are  highly  charged,  and  this  constitutes  the 
source  of  the  solar  surface  electrical  density.  Violent  readjust- 
ments are  continuously  necessary  in  the  high  levels  of  all 
atmospheres. 

Atmospheric  Electricity  and  the  Diurnal  Convection 

Similar  computations  were  applied  to  the  data  of  the  diurnal 
distribution  of  the  atmospheric  electricity,  using  the  same 
elements  that  were  described  on  pages  98  to  112  of  the  Meteoro- 
logical Treatise,  from  the  surface  up  to  3,000  meters.  Table  112, 
and  Figs.  35  and  36,  briefly  indicate  these  results.  The  figures 
are  somewhat  smoothed,  because  a  single  computation  is  not 
sufficient  to  produce  accurate  normal  values,  but  the  general 
distribution  is  clearly  indicated.  The  ionization  is  shown  on 
Fig.  35.  It  is  seen  that  we  have, 

1.  A  maximum  at  the  8  A.M.  and  8  P.M.  hours  at  the  surface, 
becoming  6  A.M.  and  10  P.M.  on  the  3,000-meter  level. 

2.  There  is  a  minimum  from  2  P.M.  at  the  surface  to  4  P.M. 
at  3,000  meters,  and  there  is  a  morning  minimum  at  about 

2A.M. 

3.  At  the   1,000-meter   level,  generally  near    the  cumulus 
cloud  base,  there  is  a  minimum,  with  negative  ionization  (ab- 
sorption).    This  leaves  positive  ionization  above  and  below, 
so  that  this  level  is  favorable  for  great  electrical  disturbances, 


ATMOSPHERIC   ELECTRICITY 


343 


as  in  the  production  of  thunderstorms,  and  lightning  discharges 
either  horizontally  or  to  the  earth. 

4.  The  lowest  curve  represents  the  surface  electric  potential, 
with  maxima  at  8  A.M.,  8  P.M.,  and  minima  at  4  A.M.,  2  P.M. 
These  must  be  closely  associated  with  the  thermodynamic 
ionization  as  computed.  The  system  of  arrows  for  the  hor- 
izontal and  vertical  electric  currents,  H,  V,  which  produce  the 
observed  diurnal  components  of  the  magnetic  field  in  Argentina 
(Meteorological  Treatise,  pages  317-329),  evidently  has  about 
the  same  distribution  with  abrupt  changes  at  8  A.M.  and  8  P.M. 
Similarly,  the  8  A.M.,  8  P.M.  maxima  of  the  diurnal  vapor 
pressure,  coefficient  of  electrical  dissipation,  evaporation,  and 
other  surface  phenomena,  must  depend  upon  this  thermodynamic 
process.  Further  researches  will  be  continued. 

TABLE  112 

THE  DIURNAL  VARIATIONS  OF  ATMOSPHERIC  ELECTRICITY 
Cordoba-Pilar  Data — Volts  per  Meter 


2 

2A.M. 

6A.M. 

10  A.M. 

2  P.M. 

6P.M. 

10  P.M. 

3000  

98  07 

*98  11 

97  99 

97  71 

97  77 

97  86 

2500  .. 

97  54 

97  43 

97  32 

97  43 

97  66 

97  43 

2000... 

96  71 

96  91 

96  90 

97  15 

97  52 

96  91 

1500 

96  80 

96  29 

96  15 

96  92 

97  21 

96  29 

1000  
800 

97.09 
96  55 

95.26 
95  78 

95.86 
95  17 

95.88 
96  27 

96.39 
96  99 

95.46 
95  65 

600... 

95  97 

95  57 

94  87 

96  69 

97  09 

95  57 

400  

95  09 

95  01 

94  96 

96  83 

96  89 

95  34 

200  

94.23 

93.82 

95.11 

96  36 

96.42 

94  85 

000... 

93.38 

92  44 

94  94 

95  49 

95  42 

94  06 

Total  Voltage  =  S  (  21-20  )  (Fi-Fo) 


3000  
2500  
2000 

1325 
1060 
645 

2081 
1742 
1481 

1341 
1006 
796 

1090 
950 
810 

977 
922 

852 

1574 
1359 
1099 

1500  .. 

690 

1171 

421 

695 

697 

789 

1000  
800  . 

835 

727 

656 
760 

276 
138 

175 
251 

287 
407 

374 
412 

600  
400  

611 
435 

718 
606 

78 
96 

335 
363 

427 
389 

396 
350 

200  
000  

263 
93 

368 
92 

126 
92 

269 
95 

295 
95 

252 
94 

344  A   TREATISE   ON   THE   SUN'S   RADIATION 

10P.M.        2A.M.  6A.M..       10A.M.         2P.M.          6P.M.          10P.M.        2A.M. 


FlG.  35.     lonization,  Potential  Energy  and  Frequency. 


ATMOSPHERIC   ELECTRICITY  345 

Table  112  and  Fig.  36  contain  the  results  of  the  compu- 
tations of  the  total  voltage  in  the  distribution  caused  by  the 
diurnal  period,  that  is,  the  solar  radiation  during  the  day  and 
the  obscuration  during  the  night.  It  is  seen  that  there  exists 
on  a  given  level  z,  generally  from  the  surface  to  3,000  meters,  a 
minimum  value  at  2  P.M.,  with  maxima  averaging  8  A.M.  and 
8  P.M.  The  computed  and  the  observed  systems  are  prac- 
tically in  accord  throughout  the  twenty-four  hours,  and  in  all 
the  levels  of  record.  There  is  an  isolated  maximum  extending 
along  near  the  1,000-meter  level,  from  10  P.M.  to  6  A.M.,  at  the 
top  of  the  level  of  vapor  convection.  The  midday  minimum 
extends  upward  toward  the  right  on  the  diagram,  and  it  gradu- 
ally spreads  out  into  a  semi-diurnal  curve  in  the  high  levels. 
This  entire  series  of  phenomena  is,  therefore,  closely  connected 
with  the  temperature  convection,  and  they  are  all  the  by- 
products of  it. 

There  have  been  many  speculative  conjectures  regarding  the 
origin  of  these  semi-diurnal  meteorological  periods,  but  they 
have  been  usually  of  a  secondary  character.  The  primary  cause 
is  clearly  to  be  ascribed  to  the  many  complex  processes  which  are 
due  to  the  thermodynamics  of  radiation.  It  is  thought  that 
with  sufficient  experience  the  formulas  that  have  been  deduced 
here,  and  illustrated,  can  be  made  to  yield  other  valuable  data 
regarding  the  atomic  and  subatomic  activities  which  are  con- 
cerned in  the  variations  of  the  fundamental  terms  expressed  by 
T\  n\  NI  hi  v\  X  and  their  very  numerous  derivatives. 

In  Figs.  35  and  36  it  is  seen  that,  by  interpolating  the  values 
of  F/meter  along  any  level,  the  well-known  distribution  is 
obtained,  with  maxima  at  8  A.M.  and  8  P.M.,  and  with  minima  at 
4  A.M.  and  2  P.M.,  especially  above  1,000  meters. 

The  Fundamental  Quantities  in  Meteorology  and  in 
Astrophysics 

It  has  become  evident  that  the  quantities  which  are  to  be 
most  useful  in  the  higher  problems  of  Meteorology  and  Astro- 
physics are  those  which  develop  the  fundamental  terms  of 


346 


A   TREATISE    ON   THE    SUN  S    RADIATION 


10  P.M.         2  A 


6  P.M.          10  P.M".          2  A.M. 


.Volts.,  300..  hf.m  10om.=  3QQ_.  g  («.-«.)    i,  100m. 
Meter        e  e         (y,-t?0),0    » 

FIG.  36.    Atmospheric  Electricity. 


FUNDAMENTAL   QUALITIES  347 

our  equations,  namely,  (T,  n,  N}  h,  k).  The  usual  forms  of 
the  Boyle-  Gay  Lussac  Law, 

(323)  P  =  PRT,  Pv  =  RT, 

P  =  —  .mR.T,  PV  =  KT 

m 

may  be  written, 

(324)  (n  k  T)  m  v  =  m  (v  n  k)  T. 

This  involves  the  following  definitions: 

Pressure.  Since  k  T  is  constant  =  3.7145  X  1(T14  (C.  G.  S.) 
earth,  and  "  "  "  =  2.9179  X  KT11. 

sun,  it  represents  the  mean  kinetic  energy  of  one  //-atom,  and 
the  pressure  per  unit  volume  is  proportional  to  the  .number  of 
//-atoms  in  the  specific  volume,  P  =  n  k  T. 

For  any  other  gas  whose  molecular  weight  is  m  the  pressure 
is  (n  k  T)  m  per  unit  volume,  and  in  the  volume  v  the  pressure 
or  kinetic  energy  is, 

(325)  P  V  =  (n  k  T)  m  v. 

Gas  Efficiency.  Grouping  the  same  quantities  in  the  second 
form  (v  n  k)  ,  k  is  now  a  variable,  and  we  have  the  gas  efficiency 
of  one  //-atom  for  one*  degree  of  temperature, 

(326)  R=  (vnk) 

so  that  the  total  kinetic  energy  for  m  atoms  and  the  tem- 
perature T,  is 

(327)  K  T  =  m  (n  v  k)  T. 
The  corresponding  dimensions  are: 


=KT. 

Hence,  the  primitive  equations  become: 

(329)  P  =  n  k  T  for  the  kinetic  energy  of  //-atoms  per  unit  v. 

(330)  P  V  =  N  k  T  for  the  kinetic  energy  of  m  and  //"-atoms  in  v. 
The  entire  thermodynamic  system,  therefore,  depends  upon 

the  variations  in  (n.  N)  and  these  are   more  fundamental  than 
(P.  p.  R.  T),  since  the  latter  are  summation  or  integral  results. 


348 


A   TREATISE   ON   THE    SUN  S    RADIATION 


Practical  Series  of,  Thermodynamic  Terms 
Since  k  T  =  constant,  n  is  immediately  found  from 
(331)  n  =  7-^  =    °,  ™    •,  from  barometric  data. 

K  J.  K  JL 

N  must  be  computed  from  the  density  by  the  non-adiabatic 
formulas.     This  can  be  shortened  somewhat,  as  follows: 


1 


The  value  of  —  = 


=  0.711  [9.85194].     Assuming   an 

initial  pressure  by  the  barometer  BQ,  and  an  initial  air  density 
PQ,  it  follows  that  the  difference  of  the  barometric  logarithms 
multiplied  by  0.711  is  the  difference  of  the  density  logarithms. 
For  an  example,  Uccle,  September  13,  1911, 


Height  2  in  meters  
B  (M.K.S.) 

4000 
0  4695 

5000 
0  4136 

6000  meters 
0  3636 

log  B 

—  1  67164 

—  1  61658 

—  1  56058 

(log  BI—  log  .BO)  

—0.05506 

—0.05600 

Factor  

0.711 

0.711 

doe:  Di  —  loe:  DO) 

—0  03915 

-0  03982 

log;  DO 

—  1  93158 

log  pi  

-1.89243 

.     —1.85261 

p  density  

0.8542 

0.7806 

0.7122 

loe;  v 

0  06842 

0  10757 

0  14739 

v  volume  

1  .  1706 

1.2811 

1.4041 

N  =  n  m  v  —  n  m/p  is  now  readily  computed. 

(334)  R  =  v  n  k  can  be  found  at  once  from  R  =  —=,  =  — Tpr. 

pi         pi 

We,  therefore,  have  obtained, 

P,  p,  v,  R,  n,  N,  at  the  temperature  T. 

k  T       constant 
Then  k  =  —=-  =  ™ — ,  the  kinetic  coefficient. 

(335)  Then  h  =  g  (zi  —  z0)  .  -       -  .  —  the  potential  coefficient. 

Z>1   —  VQ      HIQ 

This  system  is  compact  and  simple,  and  it  at  once  provides 
the  data  for  physical  studies  in  atmospheres.     The  entire  future 


ENERGIES   IN   THE   ORBIT 


349 


of  meteorology  and  astrophysics  depends  upon  obtaining  the 
correct  non-adiabatic  density  p  and  volume  v.  It  is  evident 
that  but  little  progress  has  been  made  heretofore  for  lack  of  this 
fundamental  quantity. 


The  Relations  Between  the  Kinetic  and  the  Potential  Energies  in 
Orbital  Oscillations 


Assume  the  following  data: 
VQ  =  the  velocity  of  the  mass  P  in  an  orbit  of  amplitude  a, 

with  the  angular  velocity  co  =  — -  =  20  p  in  the  periodic 

time  0. 

/  =  the  time  elapsed  from  the  epoch  0  on  the  axis  0  B. 
ti  =  the  time  elapsed  from 
the  epoch  a  on  the  axis  O  C. 
1  co 

" =  T  =  2;  =  the  fre- 

quency. 
<p  =  the  phase  angle  F=  co  (t  — 

0)  =  co  (/!  -  a). 
m  =  an  integer   number  of 

rotations. 

(336)  Distance,   x  =  a  cos  co 
(t  -  j8),  for   the   epoch  0 
and  time  /. 

(337)  Velocity.  ux  =  —  a  co  sin 
co  (/  —  j8),  along  the  axis  x. 

(338)  Acceleration.     ft  =  -L  cos  <p  =  —  a  co2  cos  co  (t  —  0), 

for  <p  =  (t  -  0). 

(339)  Kinetic  Energy.    E  =  f  m  v<?  in  the  orbit. 

Ex=  \m  co2  a2  cos2  co  (/  —  0)  = 

}  m  co2  a2  [1  +  cos  2  co  (/  -  0)]. 

JEX  ranges  from  +1  to  —  1,  and  its  integral  value  in  one 
period  vanishes. 


FIG.  37.     Potential  and  Kinetic 
Energies. 


350  A   TREATISE   ON    THE    SUN'S    RADIATION 

(340)  Average.     Ex  =  J  m  co2  a2  =  J  m  v<?  =  |  maximum  value. 
Potential  Energy.     Kinetic  Energy  +  Potential  Energy  =  a 

constant  along  the  axis  x  of  the  free  path  —  \m  v02. 

(341)  Px  +  \  m  ux2  =  \  m  vQ2  =  constant. 

(342)  Px  =  J  w  (V  -  tf,2)  =  I  m  a2  w2  [1  -  sin2  co  (/  -  >)] 

=  |  w  a2  co2  cos2  w  (J  —  )3)  = 

J  w  co2  *2. 

PS  is  a  maximum  when  Ez  =  0;  Px  =  0  when  Ex  is  a 
maximum. 

It  is  evident  that  the  periodic  oscillations  along  the  free  paths 
of  the  collisions  will  be  sorted  into  groups  according  to  the  lengths 
of  the  free  paths.  If  we  identify  Ex  with  the  thermodynamic 
h  it  is  possible  to  compute  the  correlative  quantities  in  terms 
of  the  electromagnetic  data.  The  interpenetration  of  the  orbits 
of  the  negative  electrons  which  are  carried  along  with  the  positive 
nucleus,  whose  sphere  of  action  is  the  limit  of  the  free  path, 
will  produce  subatomic  disturbances  in  the  configurations  and 
temporary  instability.  Bohr  conceives  that  the  negative  electron 
charges  e  pass  suddenly  from  one  stable  configuration  to  an- 
other in  order  to  effect  generation  of  radiation.  This  would 
render  the  atoms  excessively  unstable,  inasmuch  as  the  number 
of  collisions  and  wave  frequency  is  an  enormous  numerical 
quantity  per  second.  The  subatomic  agitations  conform  to  the 
X-ray  series  of  spectra,  and  the*  interatomic  collisions,  which 
are  external,  produce  the  visible  series  of  spectra  for  a  given 
element.  This  allows  full  scope  for  the  development  of  both 
types  of  series  in  terms  of  the  variables  h,  k. 

Bohr's  Theory  of  Non-Radiating  Orbits  in  Atoms 

Bohr's  theory  of  successive  non-radiating  orbits  in  the 
structure  of  the  atoms  of  the  chemical  elements,  in  order  to 
account  for  the  observed  position  of  the  lines  of  a  series  in  the 
spectrum,  has  attained  such  success  as  to  become  the  basis  for 
further  investigations.  It  assumes  that  the  orbital  structure 
of  the  electrons  depends  upon  the  internal  constants,  h  = 
6.548  X  10~27,  m  =  8.845  X  10~28,  e  =  4.774  X  lO'10  E.  S.  U,  or 
1.5913  X  ID'20  E.  M.  U.  Thence,  it  follows  that, 


BOHR'S   THEORY   OF   RADIATION  351 


(343) 


r  being  an  integer  characteristic  for  the  orbits,  1,  2,  3,  4  .  .  . 
For  r  =  1,  the  inner  orbit  gives  the  convergence  frequency. 

The  Series  of  Spectral  Lines 

The  frequencies  of  the  series  of  spectral  lines  can  be  expressed 
by  B  aimer's  Formula, 

(344)  r  -V(l  -  £,) 

where  m  takes  successive  integral  values,  or  by  Bohr's  Formula, 

2  7T2  w  e4  /  1         1  \ 

(345)  ^_^_  (_-_),  - 

where  T2  and  n  are  whole  numbers. 

w  =  mass,  e  =  charge,  /?  =  potential  of  the  electron,  such  that 


(346)  =  3.235  X  1015,  for  E  =  e. 

Since  the  curves  on  Fig.  32  are  functions  of  T  and  /?,  both 
being  variable,  we  may  note  the  following  fact:  If  the  h  — 
ordinates  be  drawn  for  equal  T—  intervals,  as  7\—  T0=A  degrees, 
they  are  spaced  in  positions  very  similar  to  those  of  the  above 
formulas.  It  is  supposed  that  this  is  in  harmony  with  the 
computations  which  depend  upon  h  and  T,  as  they  are  related 
to  the  thermodynamic  conditions  of  the  gas.  In  this  case  the 
spacing  depends  upon  the  temperature  of  the  live  emissions  in 
succession,  and  they  become  a  method  of  determining  the 
temperatures  of  the  solar  gases  at  different  depths.  Further- 
more, variations  in  the  positions  of  the  lines  of  a  series,  as 
hydrogen,  indicate  temporary  changes  in  the  local  temperatures 
of  solar  emission.  The  positions  of  the  lines,  as  measured  in 
the  spectrum,  may  be  used  inversely  to  establish  the  position 
and  the  correct  curve  of  the  (h  .  T)  function. 

It  is  evident  that  h  is  a  variable,  and  we  shall  show  that  the 
spectrum  lines  can  be  produced  by  the  vibrations  imposed  upon 
the  electronic  orbits  by  reaction  from  the  external  collisions  of  the 
atoms  and  molecules. 


352  A   TREATISE   ON    THE    SUN'S    RADIATION 


Derivation  of  the  Orbital  Formulas 

Assume  the  following  relations: 
a  =  the  orbital  radius. 
v  =  the  orbital  frequency  =  the  vibration  frequency. 

/> 

V  —  —  =  the  potential  at  the  distance  r. 

3  V       e 

(347)  £=-——  =  --  =  the  central  force. 

(j  V  ¥ 

E  e 

(348)  W  =  —  =  V  e  =  the  work  done  between  charges  E,  e. 

v 

(349)  co  =  -  -  =  2irv  =  2ir  —  =the  angular  velocity. 

6  A 

(350)  e  =  -  -  =  —  =  —  =  the  periodic  time. 

CO  V  C 

1  co          c  v          mo2 

(351)  ,  =  T  =  —  =  -=—  =  — . 

(352)  <p  =  w  t  =  --  .t=2irv.t=—  -  .  /,  for  the  elapsed  time  t. 

6  A 

e  E 

(353)  V  =  = ,  if  the  central  charge  is  E  =  n  e. 

ra        rna 

From  the  orbital  velocity  v,  the  constant  kinetic  energy  in  the 
orbit  y2  m  u2,  and  the  central  acceleration  /,  have  the  following 
formulas : 

(354)  Velocity,     v  =  2ir  a  .  i>  =  a  u  =  2ir  a  . -^- =  ^f- 

A  r  n 

E  e       e  e 

(355)  Kinetic  Energy.    %miP  =  hv=V  e  = =  —  = 

rna      ra 

g  (zi  —  ZQ) 


,..  mv2       2hv      2V  e       2Ee 

(356)  Acceleration  to  Center.     = = = ; 

a  a  a          rna2 


BIGELOW'S   AND    BOHR'S   FREQUENCY  353 

r  h2 

(357)  Radius  of  Orbit,      a  =  — .     Log.  2  7r2  m  e2  = 

&  7T    ttt  C 

-  45.59979. 


(358)  Velocity.       »  =  — T-.        Log.    2 ire2       =  -   18.15594. 

2  7T2  fH  6* 

(359)  Frequency.    v  =      ^  ^  2ir2me*    =  -  64.95755. 

47T3  w  e4 

(360)  Angular  Velocity,     co  =  — r/J~-    L°S-  4  7r3  w  g4  = 

-  63.75573. 

9      2  AM  /?3 

(361)  Potential.     F=     ^2     .     Log.  2Tr2me*  =  -  54.27867. 

M7  7J"  /7   77   T 

(362)  Nuclear  charge.     £  =  - 


2  7T2  m  & 
(363)  Kinetic  Energy.    J  m  ^2  =  — ^r^— .    Log.  2  7r2  m 


-  64.95755. 


N 

(364)  Acceleration  to  Center.     -  = 


. 

d  T      ft 

Log.    8  r4  w2  e6  =  -  108.85837. 

These  can  be  used  in  computing  the  elements  in  the  jRT-series, 
Z-series,  and  the  other  constituents  of  the  spectrum. 

From  the  theory  of  electronic  orbits  we  have  directly  the 

centrifugal  force  =  %mv2  =  hv  =  -—  ,  where  E  is  the  charge 

on  the  positive  nucleus,  and  n  the  number  of  electrons;  hydro- 
gen n  =  2.  For  the  velocity  in  the  orbit,  v  =  2  TT  a.  v,  where 
the  angular  frequency  and  the  number  of  vibrations  v  are  assumed 
to  be  the  same.  Hence,  we  have, 


(365)  Frequency.  „  =          = 

(366)  Radius.  a 


mirv 


354 


A   TREATISE    ON    THE    SUN?S    RADIATION 


The  central  force  of  acceleration  is, 

mv2       2Ee      Ee   . 
(367)       /  = =  — £-  =  — ,  for  hydrogen,  n  =  2. 

The  charge  on  the  central  nucleus  of  positive  electricity  is, 


Charge  (+),£  = 


m  a 


mi) 


h 


=  —  =  e.     Hence, 

e  TT 


(368) 


m 


=  -TT-.     Thence, 


,       N  . 

(369)  Frequency,  *  = 


-i-2),  hydrogen. 


We  compute  the  two  formulas  for  v  and  X  =  — : 


v        g  (zi  - 

zo)       1        1 

V  = 

2  w*  m  e*   1  1  \ 

Lyman 
as  observed, 
also 
Millikan 

Science 
April  6, 
1917 

0.00003650 

(vi  —  wo)  10  *  wio  *   h 
Bigelow 

Bohr 

Log.  g  (ZI-ZG) 

Log.  (Vi—Vo)w 

Log.  nw 
Log.  h 

Log.  ,w 
Log.  c 

Log.  X™ 
Wave  length 
(cm)  Xm 

13.13805 

5.15619 
17.11295 
-25.96720 

Log.   7T2 

Log.  m 
Log.  e* 

Log.  2 
Log.  A' 

Log.  j/ 
Log.  c 

Log.  X 
X 

0.99430 
-28.94670 
-38.71552 

-64.65652 

0.30103 
-79.44770 

-  2.23634 

14.90171 
10.47712 

-79.74873 

14.90779 
10.47712 

-  5.57541 
0.00003760 

-  5.56933 
0.00003710 

The  Bigelow  formula  seems  to  be  equivalent  to  the  Bohr 
formula  for  the  case  of  the  hydrogen  series,  since  the  convergence 
wave  length  is  practically  the  same.  It  is  evident  that  since  in 
the  Bigelow  formula  the  h  is  an  external  potential,  while  in  the 
Bohr  formula  h  is  a  function  of  the  kinetic  energy, 


h  =     m 


MOSELEY'S  LAW  355 

it  will  be  necessary  to  examine  the  formulas  in  their  other 
relations. 

Our  thermodynamic  data  refer  in  all  cases  to  the  maximum 

2891 
vm,  Am,  in  the  Wien  Displacement  Law,  Am  =  -7^—,  so  that 


m 


these  results  are  practically  identical.  It  should  be  noted  that 
the  Bigelow  form  uses  the  hm  of  the  external  potential,  while 
Bohr  uses  //  =  the  constant  or  adiabatic  value.  We  may, 
therefore,  conclude  that  the  Bohr  computation  refers  to  the 
adiabatic  case,  generally,  and  the  Bigelow  computation  to 
the  nonadiabatic  case  near  the  bottom  of  the  solar  reversing 
layer. 

Moseley's  equation  for  the  relation  between  the  frequency 
and  the  atomic  numbers, 

(370)  vN  =  A(N-  b)2  or  VN  =  (a  +  b  N)2 

or  Uhler's  more  accurate  equation,  which  is  hyperbolic, 

(371)  vN  =  A+BN  +  D6_N  (Physical  Review,  April,  1917), 

* 

shows  that  the  ultimate  relations  between  the  atomic  numbers 
in  the  Ka,  K^  La,  L&  Ly,  types  of  the  X-ray  series  are  really 
very  complicated.  It  remains  to  be  seen  what  form  further 
experiments  will  assign  to  these  structural  relations.  We  shall, 
therefore,  reserve  this  subject  for  further  examination,  though 
it  has  seemed  proper  in  this  chapter  to  indicate  some  of  the 
interesting  developments  which  come  from  the  non-adiabatic 
thermodynamics  . 

Sanford  applies  the  Bohr  theory  'to  the  K  and  L  series  with 
much  success  in  Physical  Review,  May,  1917.  These  subjects 
will  be  resumed  in  a  further  publication  on  the  structure  of 
matter. 

Moseley's  Law 

Moseley  found  an  important  relation  between  the  atomic 
numbers  of  the  elements  in  certain'  series,  by  assuming  that  the 


356  A   TREATISE   ON   THE    SUN*S   RADIATION 

addition  of  the  b  unit  charges,  b  .  e,  to  the  nucleus  suffices  to  pass 
from  one  element  N  to  the  next  in  order  N+b.  For  r  =  1 
(convergence) . 

•in    *"'"  'in 

(372)  /  =  ~  =  ^-  .  4  7r2a2  »2  =  4  7r2  m  a  ,2 

^Q7cA     /        2£]g     £i«      j  /        E2e 

(373)  /i  = = — -  and/2  =  — r,  for  w  =  2.  hence. 

w  «i2       ai2  #22 

(374)  /i  ai2  =  Ej.  e  =  4  ?r2 

(375)  /2a22  =  E2e  =  4 


Moseley  finds  that  the  frequency  is  proportional  to  the 
square  of  the  nuclear  charge. 

(376)  —  =  TTg  =  Moseley's  Law.     Hence, 

(377)  f1  =  f^4  .  —^  and  £i3  ^3  =  £23  a23,  so  that 

£Lz          ±Lz        #2 

(378)  (Millikan)     EI  EI  =  a2  E2. 

This  is  derived  from  the  central  acceleration.  On  the  other 
hand,  from  equation  (330),  we  have,  by  assuming  that  the  kinetic 
energy  in  the  orbit  is  a  constant,  a  different  relation. 

(379)  (Bigelow)    ai  E2  =  a2  EI 

There  are,  therefore,  two  distinct  solutions  by  (346)  and 
(347),  the  former  depending  upon  assuming  that  the  frequency 
in  the  orbits  of  the  electrons  is  the  same  as  the  frequency  in  the 
spectral  lines;  the  latter  assumes  that  the  kinetic  energy  in  the 
orbit  is  a  constant  for  all  electrons,  the  radial  distance  conform- 
ing thereto.  The  former  assumes  that  h  is  constant,  and  the 
latter  that  h  is  a  variable;  the  former  supposes  that  the  funda- 
mental relations  are  internal  to  the  atom,  the  latter  that  the 
primary  relations  are  external  to  the  atom,  and  that  the  varia- 
tions in  the  position  of  the  lines  of  the  spectrum  are  due  to 
perturbations  impressed  by  interpenetration  of  the  active  orbits 
of  the  electric  charges  during  the  collisions.  The  Bohr  theory 
makes  the  variation  of  energy  which  causes  radiation  to  depend 


EVALUATION   OF   SERIES   FACTORS 


357 


TABLE  112 


. 

Evaluation  of 


(I         1\ 
(-2  --,  j 


l         !v 

^  -  -2j 


J__JL 

cos<f> 

4> 

'  -p^^^ 

Tl2        T22 

"""-V^^ 

Tl  =    1 

T2  =  2 

0.750 

0.750 

41°     25' 

X. 

3 

0.889 

0.889 

27       15 

N. 

4 

0.938 

0.938 

20       17 

5 

0.960 

0.960 

16       16 

M 

6 

0.972 

0.972 

13       36 

jb 

7 

0.980 

0.980 

11       15 

1 

8 

0.984 

0.984 

10       16 

3' 

9 

0.988 

0.988 

8       52 

3 

10 

0.990 

0.990 

8        8 

11 

0.992 

0.992 

7      15 

Conver 

gence  v 

1.000 

\ 

n-2 

0.139 

0.556 

56°     12' 

11     7    5   - 

132                       *ij 
H 

Ix 

4 

0.188 

0.752 

41       13 

X 

5 

0.210 

0.840 

32      51 

bd 

\ 

6 

0.222 

0.888 

27      23 

& 

\ 

7 

0.230 

0.920 

23         5 

c 

\ 

8 

0.234 

0.936 

20      46 

0 

\ 

9 

0.238 

0.952 

17       50 

n 

V 

10 

0.240 

0.960 

16         7 

\ 

11 

0.242 

0.968 

14       31 

\ 

Conver 

gence  v 

0.250 

11 

-^ 

8651             3        "Tj 

O 

sr 

*SN. 

Tl  =  3 

T2  =  4 

0.049 

0.441 

65°     50' 

jjljs 

\ 

5 

0.071 

0.640 

50        8 

6 

0.083 

0.748 

41       35 

& 

7 

0.091 

0.820 

34       28 

1 

8 

0.096 

0.865 

30         8 

9 

0.099 

0.892 

26       52 

n 

k 

10 

0.101 

0.910 

24       30 

V 

11 

0.103 

0.928 

21       52 

\ 

Conver 

gence  v 

0.111 

11    9    7      6       5        4    £ 

(6 

FIG.  38.     Orbital  Distributions. 

358  A   TREATISE   ON   THE   SUN'S   RADIATION 

upon  the  electrons  jumping  from  one  stable  non-radiating  orbit 
to  another,  but  this  would  make  the  structure  of  atoms  wholly 
unstable  and  precarious.  The  flat  Saturnian  systems  of  orbits 
should  probably  be  superseded  by  a  series  of  orbits  which  are 
arranged  upon  the  surface  of  a  sphere,  in  order  to  produce  the 
polarizations  and  magnetizations  that  exist  in  molecules. 

The  evaluation  of  the  term  f  — -Jin  the  expression  for 

the  frequency  v  gives  the  relative  position  of  the  spectrum  lines 
in  different  series.  The  function  cos  <p  is  formed  from  successive 
divisions  by  the  convergence  value,  computed  for  TZ  =  °°,  and  <f> 
is  the  angle  from  the  equator.  If  the  electrons  should  revolve 
on  these  planes,  as  indicated,  the  frequencies  would  correspond 
with  an  Amperean  polarized  sphere.  If  the  several  atoms  occur 
in  different  environments  of  h,  depths,  densities,  temperatures, 
the  corresponding  radiations  conform  to  those  observed  in  the 
spectrum.  This  research  will  be  continued 

The  Electronic  Orbits  in  the  K  and  L  Series  of  Radiation  Lines 

for  h- Variable. 

We  have  computed  the  values  of  the  several  terms  according 
to  formulas  (322)  to  (339)  for  the  Ka  and  La  series,  quoting 
Uhler's  and  Sanford's  wave  length  values  X,  as  given  in  the 
American  Physical  Review  for  April  and  May,  1917.  From  these 
values  of  X  the  frequency  v  is  computed,  and  with  this  v  the 
corresponding  variable  h  by  the  formula, 

(380)  '        <<1M 


This  variable  h  is  then  applied  to  a,  v,  V,  %  m  v2, ,  E,  in 

succession.     Finally,  we  have, 

v> 

(381)  A  =  —  =  a  constant  by  Moseley's  Law. 

It  is  noted  that  in  the  Ka  radiation  the  value  of  A  is  somewhat 
smaller  in  the  middle  than  at  the  ends  of  the  series;  the  same  is 
true  of  the  La  series.  In  each  case  the  divergence  is  small,  and 


EVALUATION   OF   SERIES   FACTORS 


359 


^ 


2    ^ 
< 

s 

<        I 

& 


> 


rH         00  IO  C-0000          rH  I 


ooooo 


iocoeoc~rH  oseocoorH  t-ooi 

OS  00  O  Tj<  IO  OJlOOrHCO  O  rH  rH  1 

t-tooooooo  oooooooooo  ooso< 

ooooo  ooooo  ooo< 

t-t-t-c-t-  t-t-t-t-t-  t- t- t- I 


o     ososososos 
f      Mill 


TjICOTf  t-  OS 

1000  co  10 1- 
oooo  os  os  os 

ososososos 
Mill 


« t-oioc- 

)OS  rHCOTjl 
<rHCMCMCM 


oooooooooo     oooooooooo     oooooooooo 
Mill       I   I   I   II       I   I   I   I   I 


os  10 1-  co  co 

OS^J«t-rH-^ 

«O  t- t-OOOO 


IS  §2; 


00  OS  OS  03  OS 


ososososos     ososososos      ososososos 


•  t-  CO  t-  rH 


rH  01  CO  •*  TH 

ddddd 


>  NO]  IOOO  rH 

>  OS  N  OS  <£>  O3 
rHIOOSCOrH          IO  OS  IN  OS  CM 


OOOOO       Oi 


T-H  ioo]  i 
oooo  os 


loeocoioio      t-iot-ioio      t-os>OrH( 
CM  c~  o  01  os      COIOOSCON      os^rHooi 


rH          ®SSrHrH 

7  77777 


Mill 


ososososos     ososososos     ososososos 
Mill       Mill       Mill 


t-eo  r-i  t- 

t-  ION  i-H  ( 


.OSOOO 

i  r-i  d  d  d 


OS^DOSrHO          COOOrHCOO 

rHrHCMNCO         CO  CO  CO  5<  "         TT  ^J< -^  IO  IO         »O  US  «O  fO  CO 

oodo'd     o'dodo'     oo'do'd     do  odd 


Vfr"^ 

t-OSrHkO         !>OSrHTj<CO 


f(N 

'« 

00   CO  O  Ci  O  C1 


!§S§ 


OO  lOrHOO  IO 

os  os  os  os  os 


lOrHCO  t-< 

OOOSrHCOt-         rHTj«t-lO< 
)0< 


CMioosrHeo      ^lococoio      «OT)«eo< 

OOOSOCMCO        -^<iOCOt-OS        OrHCM^Tf 
rHrHCMCSJCM        CM  CM  CM  CM  CO        CO  CO  CO  CO  CO 


ososososos     ososososos     ososososos 


t-  TflO 
OS  OS  O 


00   OOOOO 

i  77777 


ooooo 

77777 


ooooo  ooooo  ooooo 


co      ioiOTj<  o< 

.rH         OO  IOCO  rH  ( 


CjJNCMCjICH       CjJCjlCqCUCjl       CO  Cjl  «N  Cjl  <N        Cjl  CM  Cjl  <N  <N 


T-l  IO  T-l  Tjl  00 

JO££CO<£> 
OS  OS  CO  T-l  iH 


rHOsoot-«o     iOTj«cocoeo 


CM  N  CM  CM  r-J 


OOOIOOOS 
OS  kO  T}<  rj<  »O 


360 


If 

^ 


3     » 

r,i       ^ 


A   TREATISE   ON   THE   SUN'S  RADIATION 

CD  rji  t-l  Hi  t~        CSOOOOCS^  CJCOOOW5CO        t-COt>Tii^-l  IO 

t- Tfi  lO  O  t-        Ot-Ni-HOO  t-COCOTfCM        ^OOOCOCO  CM 

-t-  OO'O  OTH         ^Hi-Hi-IOlCO  kfl-^i-KMCO         I00£t>t>  10 


(N 


-si    « 


«     8 

tn 


OS-^OOC^ 
OCOOSiH 


(COCO        CO 


^oooooooooo     oooooooooo     oooooooooo     oooooooooo     oo 
I    I    I    I    I        I    I    I    I    I        I    I    I    I    I        I    I    I    I    I        I 


ICM  CO  t-  M  t-  CO  O  i-<  CO  CO  CO 

•rH  TjlOOOiHOS  00  1O  rH  CO  "3 

I  •**  COCOCMCOCO  •»*  CM  00  t- t- 

ICO  ^WCOt-OO  0>OO^CO 


b-COOiiOCO 
t-  CO  (M  00  O 

C7i  Ci  O  O  O 


Ilii 

i-t  CM  CO  ' 

iom  m  ie>  10 


Mill       I   I    I   II       I    I    I    I    I 


S^:! 


UOOOJOWO       CDt^OJOi-H        COTtH«COO>        OiOCMCO'i*        t- 
>COCOt^t-t-        t-t~t-0000        OOOOOOOOOO       OOOSO5OOJ       O 

o'odo'o'     o'o'o'o'd     o'ddo'd     do  odd     »-! 


3  00  C<J  00  i-H         COTHt-COOO         IO 

jco-*-*co      cot-t-oooo      w 

4^^4^4^  ^  ^4  ^  ^  ^  LQ 


lOtNt-Tf 

loot-koco      tHOoot^-co     ^<  co  co  I-H  66     t^cisuj-^co     o 

>050S0505        OSOSOOOOOO       OOOOOOOOt-        t-t-C-t-t>       CC 


OU5  Tj<  r-l 

COU5O  rH 
"500 

s  oo  t- 


i-l       i-lTjiOt~O        U3rHt>COCO 


o  < 


IflTjtTjicOCM        CMiHiHOOJ        OOOOt-t-CO       O 

t-t-t-t-t-     t-t-t-t-co     cococococo     co 

TTTTT  ^TTTT  ^TT'T  T 


do  odd  do  odd  ddoo'd  o'do'o'd  d 


lOCOt-OOOi        Ot-H 


Ot-HIMCOiO        CO  t-  00  OS  O 

loiomioio      U3io»oioco 


EVALUATION    OF   SERIES   FACTORS 


361 


'-teoNW  •««  us  N  CM  t-  oo  m  eo  <N  o  ocoweoo  «o  t~  os  t-  «o  Ti«t-<oa»o 

-<J<  i-l  iH  «D  5O1OOOOOOS  COCJOSeoW  <NOOOSO<O  Nt~t-Or-<  NOO(OOS«O 

sllls  §§111  §§!§§  §§§!§  Sslll  illll 

'  t-  1-  t-  t-  1-  t-  1-  t-  t-'  t-  t-  t-  t>  t-  1- 


t-  1-  t-  1-  1-      t>  t-  c- 


o  IH  «N  «N  <N  <N 


Illll 


oosoosc 
Illll 


oosooo      oawaa      ooooo 
Illll       I    I    II    I       Illll       Illll 


OOOSt-W5(M        eOCO< 

LO  Tj«  N  l-H  00          OOrH< 


><MO«OTI<     -it-«NOO( 

1S§§§    g§-S: 


t-ooo>     O» 


oooooooooo     ooooooooos 


oo  eo  1-1  •")"  < 

Mr}<  00  rH  10  < 

Oo  eo  t-  eo  > 

,_)  IO  IO  IO<0< 


10-H«<OC<I«H 
OStNOCOOO 
<O  t-OOOOOi 

OOOOO        OOOOO 


t- as  com  as  0j,_i^Ht-i».  eoiocot-t-.  i-it-rnt-i 

i-HCNiOOON  -*NOt-t-  U5<OrHCOTH  d  t- (M  ^H  < 

lOt-oscNco  coio-^OrH  t-iON<oio  r-(ioeooo< 

O  tN  ^  OS  ^H  COt-rHrHOJ  lOt-OSOCN  -^<lOt-tO< 

oooo>-i  t-i.-iMeoeo  eoeow*^1  ^J<  ^c  ••»  »o  < 


Q  t*  t-  oo  oo  oo 
^  /— *  ^—^  ^^ .— >  t~ * 


gssr( 

lOO  >-l  •,     - 

t-OSM^rH 

OO  00  OS  OS  O 


'OOOOO 

77777 


ooooos 

7777  ' 


ISSS! 

05  OS  OS  < 

I  I  I  I 


'  '  '  '  '    '  '  '  '  '    '  '  '  ' 


OOrHW< 

d  o  d  d  d 


t-  co  to  com 


1-1  <N  N  (N  W 

o  o  o  o  d 


OOOOO   OOOOO   OOOOO   OOOOO 


t-  r-tOOO  t- 


ot-OOOO 
OOO^-Hr-l 

os  oi  as  oi  a> 


«N<N<N     c; 

lOsosoj     ososososos 


csascsosos 


loeorHiovi  rHioeoioco  «-tiocot- 

OSNOOCOV5  OOIMNOOO  OOt-i-HOS 

<o  o  eo  I-H  o>  *-it~?oeooo  tomtnco 

t-  eo  Ti<  I-H  o>  oo^c<iT-(eo  cvii-Hooo 


>«0          OOrH< 


o  o  o o 


Illll  I  II  I  I  Illll  Illll  Illll  Illll 


;ssii 


>NrHNrH 

>U3§?^ 


^S§o§3    S§S! 

'^^^^^ 

I  I  I  I 


1 01 0404  ( 

I  I  I  I 


.^coc,  Nt-o^a  S5°wo< 

-•(N»Heo  —i  o  1-.  w  ^  io< 

>iovo-*feo  eoMr-ioooo  t-t^yssbio     ibi 

IOSOSOSOS  OSOSOSOOOO  OOOOOOOOOO       OOOOOOOOOO 

oooooooooo  oooooooooo 

(NOJNCOtN  N(N(N«NM  

Illll  Illll  Illll  Illll 


OOSOOW        OrH 

t-rmot-     ^t- 


Tf  O 

»-ios 


>  t-  rf  t-  i-H 

IrHl-HOSOJ 

«  -H  i-!  d  d 


362 


A   TREATISE   ON   THE    SUN'S    RADIATION 


it  proves  that  Moseley's  Law  conforms  to  the  variable  h  in  the  series. 
Since  the  potential  energy  h  represents  the  resistance  to  the  kinetic 
energy  in  the  medium,  because  we  have  assumed  the  kinetic 
energy  constant,  and  the  central  force  variable  from  one  element 
to  another,  it  is  seen  that  the  theory  of  orbits  takes  on  very 
much  greater  flexibility,  and  really  represents  the  complex  of 
the  thermodynamic  conditions  which  produce  radiation.  These 
vary  with  the  depth  in  gases,  temperature,  density,  pressure, 


Kinetic 


FlG.  39.     Interpenetration  of  the  Orbits  of  the  Electrons  in  the  Atoms 
During  Collisions. 

gas  efficiency,  surrounding  each  atom  as  represented  by  n  and  N. 
Hence,  different  subordinate  series  of  the  same  element  originate 
in  different  environments  on  the  outside,  while  the  several 
chemical  elements  are  derived  from  the  different  structural  con- 
figurations of  the  electrons  on  the  spherical  surface  of  the  atoms. 

The  Interpenetration  of  the  Electron  Orbits  at  the  Contact  of 

Collision 

The  complex  systems  of  atoms  in  collision  must  cause  inter- 
penetration  of  the  individual  orbits  of  the  electrons,  and  this 


VARIABLE    INTENSITY   OF   THE    SOLAR   RADIATION  363 

must  communicate  a  series  of  internal  vibrations  to  the  radiat- 
ing particles  or  electric  charges. 

The  potential  energy  //  in  the  radiation  function  is  related  to 
the  kinetic  energy  k  in  much  the  same  way  that  has  just  been 
described.  If  k  T  is  a  constant  then  h  v  is  a  variable,  and  a  very 
complex  variable  whose  mean  values  alone  can  appear  in  the 
thermodynamic  formulas.  The  important  point  to  recognize  is 
that  the  electrons  may  remain  on  their  stationary  orbits  accord- 
ing to  the  structure  of  the  chemical  elements,  and  acquire  the 
perturbations  producing  the  radiation  during  the  confusion  of 
interpenetration.  This  avoids  the  difficulty  in  Bohr's  theory 
which  requires  the  electrons  to  pass  from  one  orbit  to  another 
during  radiation.  The  amperean  spherical  distribution  of  the 
interacting  elementary  charges,  E  at  the  center,  and  e  in  the 
several  orbits,  probably  present  the  basis  for  the  structural 
permanences  which  are  inherent  in  the  chemical  elements. 
Fig.  39  represents  the  collision  of  two  polarized  atoms  whose 
orbits  are  projected  on  the  equatorial  plane. 

The  Electromagnetic  Waves  Due  to  the  Sudden  Motion  and  Stoppage 
of  an  Electric  Charge  in  Collisions 

Following  Heaviside's  exposition  of  the  effect  of  suddenly 
starting  or  stopping  the  motion  of  an  electric  charge,  as  in  col- 
lisions, we  have  the  distribution  of  the  electric  disturbance  D 
and  the  magnetic  induction  B,  in  producing  the  plane  elec- 
tromagnetic waves  of  Fig.  40. 

Let  p  take  on  suddenly  the  velocity  v,  so  that  in  the  time 
t  it  reaches  the  polar  position  of  the  sphere  +p.  The  outside 
radial  displacement  changes  into  a  current  along  the  meridians 
from  the  positive  to  the  negative  pole,  with  its  magnetic  induction 
on  the  parallels  in  the  vector  sense,  so  that  the  polar  field  be- 
comes the  plane  wave  [D  .  B],  as  the  sphere  enlarges.  Let  p 
be  suddenly  stopped,  then  the  displacement  reverses  from  the 
negative  to  the  positive  pole  along  the  meridians,  with  induction 
on  the  parallels  in  the  opposite  direction,  and  renewed  inner 
radial  displacement  to  the  center.  This;  also,  releases  a  plane 


364 


A   TREATISE   ON   THE    SUN'S    RADIATION 


wave  [D .  B]  at  the  positive  pole.  Similarly,  during  collisions 
of  two  orbit-atoms,  there  are  sudden  motions  and  stoppages, 
with  reversed  displacements  and  inductions,  sending  plane 


Sudden  Stopping.  Sudden  Starting. 

FIG.  40.     The  Sudden  Motion  and  Stoppage  of  an  Electric  Charge  p. 

electromagnetic  waves  into  space.  When  the  orbits  of  atoms 
in  collision  interpenetrate,  these  waves  are  complicated  in  their 
frequencies,  in  accordance  with  the  results  seen  in  their  charac- 
teristic spectrum  lines. 

Furthermore,  during  successive  collisions  at  the  end  of  the 
free  path  C,  there  exist  the  potential  and  the  kinetic  energies, 
which  may  be  analyzed  as  potential  along  the  free  path  with 
h  as  the  average  value,  and  2  h  as  the  maximum  value,  the  time 

of  one  period  being  —  second  of  time,  while  the  kinetic  energy 
v 

is  expressed  by  the  motion  in  the  circle,  such  that  k  T  =  J  m  v2. 

(382)  divW  =  -  e<>J  -  hoG  +  Q  +  j  +  H 

(383)  W  =  V  (E  -  eo)  (H  -  fc)  =  V  El  H±  =  v  (j  +  H). 


All  these  forces,  expressed  and  implied,  are  in  action  during 
the  collisions  of  complex  atoms  and  molecules. 


VARIABLE   INTENSITY   OF   THE    SOLAR    RADIATION 


365 


The  Variable  Intensity  of  the  Sun's  Radiation  in  the  26.68-Day 
Period  of  the  Synodic  Rotation 

Besides  showing  that  the  intensity  of  the  solar  radiation  is 
variable  in  the  n-year  period  to  the  amount  of  i%  to  2%,  it 
appears  that  a  similar  variation  occurs  as  the  sun  turns  on  its 
axis.  Fig.  41  gives  the  normal  direct  curves  of  the  solar  varia- 
tion, as  registered  in  the  terrestrial  magnetic  field  and  the 
meteorological  elements,  according  to  the  author's  papers  of 
1893,  1895,  1898,  and  the  Meteorological  Treatise,  1915.  The 
Cordoba-Pilar  curve  of  the  pyrheliometric  mean  intensities, 
1912-1916,  results  in  a  nearly  identical  curve,  and  this  proves 
that  the  solar  radiation  is  variable  in  solar  longitude  and  affects 
all  the  terrestrial  elements  in  the  26.68-day  period.  Similarly, 
the  La  Quiaca  pyrheliometric  data  produce  the  small  curve  in  the 


12345678    9  10  H  12  13  14  15  1617  18  19  20  21  22  23  2426  2627 


FIG.  41. — The  variable  intensity  of  the  solar  radiation  in  the  26.68-day 
synodic  period. 

inverse  for^n,  the  amplitudes  being  much  larger.  It  seems  that 
there  are  inversion  and  damping  of  the  variations  in  the  intensity 
of  the  radiation  in  the  lower  atmosphere,  a  subject  of  importance 
for  further  research. 


366  A    TREATISE   ON   THE    SUN'S    RADIATION 

International  Character  Numbers 

The  character  numbers  of  the  amplitude  of  the  disturbance 
of  the  magnetic  field,  as  published  by  the  International  Com- 
mission for  35  stations  during  1915,  produces  nearly  the  same 
inverse  curve.  This  curve  distributes  its  minor  crests  on  four 


10. 


20 


26 


FlG.  42  —Magnetic  Character  Numbers,  1915. 
International  Commission,  35  Stations. 

axes,  whose  center  is  eccentric  to  the  axis  of  rotation,  as  if  the 
center  of  radiation  revolves  about  the  center  of  mass  of  the 
sun.  This  eccentricity  of  radiation  is  evidently  sufficient  to 
form  the  basis  of  weather  forecasts,  because  these  maximum 
points  have  been  proved  to  persist  through  70  years,  1840-1916. 
It  will  be  necessary  to  abandon  the  practise  of  publishing 
magnetic  and  meteorological  data  on  the  calendar  months, 
which  has  no  scientific  meaning,  and  substitute  the  26.68-day 
period  with  epoch,  June  13.72,  1887.  Compare  the  Meteor- 
ological Treatise,  p.  334. 


VARIABLE    INTENSITY    OF   THE    SOLAR   RADIATION  367 

365-day  period  Jan.     Feb.  Mar.  April  May  Jane  July  Aug.  Sept.   Oct.   Nov.    Dec. 


Radiation 


Gr.  Cal. 
Cm.2  Min. 
4.000 


Clear 
days 


Hazy 


3.950 


days 


3.900 


Amplitude  - 


\ 


Variations  -70 


50 


\ 


Frequency 
Magnetic 


-12 


Direct 
Type 


-16 


FlG.  43. — Synchronous  variations   of  the  radiation  and  the  magnetic  field 
in  the  365-day  period. 


FIG.  44. — Synchronous  variation  of  the  radiation  and  the  magnetic  field 
in  the  20 .68-day  period. 


368  A   TREATISE   ON   THE    SUN'S    RADIATION 

Synchronism  between  the  solar  radiation  intensity  and  the  terrestrial 
magnetic  variations  in  the  jdj-day  and  the  26.68-day  periods 

The  365-day  period  is  for  the  annual  revolution  of  the  earth 
about  the  sun,  and  the  26.68-day  period  is  for  the  synodic  rota- 
tion of  the  sun  on  its  axis.  The  pyrheliometer  observations  in 
Cordoba-Pilar  have  been  divided  into  three  classes,  correspond- 
ing with  the  local  weather  conditions: 

Clear j  when  the  sky  is  apparently  free  from  haze  and  dust. 

Haze,  when  there  is  a  milky  haze  due  to  vapor  or  ice. 

Dust,  when  the  low  level  dust  has  been  raised  by  wind. 

It  was  found  practical  to  determine  correcting  ordinates 
which  should  reduce  the  haze  and  dust  days  to  clear  days.  In 
order  to  eliminate  the  effect  of  the  annual  revolution,  the  clear, 
haze,  and  dust  days  have  been  reduced  to  the  mean  clear  days, 
whose  average  value  of  the  solar  radiation  is  3.980  gr.  cal./cm.2 
min.  Fig.  43  shows  that  the  clear-day  radiation,  the  hazy-day 
radiation,  the  amplitudes  of  irregularities,  and  the  frequency  of 
the  magnetic  direct  type  have  maxima  when  the  sun  is  on  the 
earth's  equator.  Compare  Bulletin  No.  21,  U.  S.  Weather 
Bureau,  1898,  pages  100-108. 

Fig.  44  shows  that  the  solar  radiation  and  the  magnetic  field 
synchronize  in  the  26.68-day  period,  when  the  radiation  is  re- 
duced to  the  mean  clear  standard. 

It  follows  that  the  compilation  of  tables  in  the  unscientific 
calendar  months  must  be  abandoned  and  collections  in  the 
26.68-day  period  substituted.  All  observations  of  radiation 
must  be  freed  from  the  imperfections  due  to  haze  and  dust  in  the 
low  levels  before  they  are  compiled.  All  the  data  of  meteorology, 
terrestrial  magnetism,  atmospheric  electricity,  and  solar  physics 
generally,  must  conform  to  these  criteria. 

In  order  to  obtain  the  foregoing  results  it  has  been  neces- 
sary to  supplement  the  ordinary  method  of  using  the  Bouguer 
formula  by  the  table  on  page  370  of  correction- ordinates  com- 
piled at  Cordoba-Pilar,  Table  115.  They  are  the  result  of 
separating  the  5 -years'  record,  1912—1916,  into  the  three  groups 
indicated.  Other  stations  must  prepare  similar  tables  repre- 
senting their  climatic  conditions. 


EFFECT  OF  DUST  369 


The  Effect  of  Dust  in  the  Lower  Strata 

The  results  of  observations  with  pyrheliometers,  to  determine 
the  intensity  of  the  solar  radiation,  and  the  amount  of  its  varia- 
tions in  different  intervals,  have  been  always  unsatisfactory, 
because  they  were  defective  in  two  particulars:  (1)  The  poten- 
tial energy  of  the  radiation  is  omitted  entirely  in  the  computa- 
tions; (2)  the  effect  of  the  dust  and  impurities  in  the  lower 
strata  has  not  been  eliminated.  Unfortunately,  the  records  as 
usually  published  give  insufficient  data  from  which  either  of 
these  terms  can  be  fully  computed.  It  is  necessary  to  know  the 
intensity  at  various  zenith  distances,  and  the  coefficient  of  trans- 
mission; also,  the  temperature,  vapor  pressure  and  barometric 
pressure,  for  computing  the  potential  terms;  the  state  of  the  sky, 
clear,  hazy,  dusty,  force  of  the  wind,  for  determining  the  dust 
effect  on  the  scattering.  Neither  the  Smithsonian  Astrophysical 
Observatory,  nor  the  U.  S.  Weather  Bureau,  publish  their  in- 
tensities with  enough  auxiliary  data  to  make  it  practical  to 
include  their  numerous  observations  in  a  definitive  reduction, 
such  as  Cordoba-Pttar  now  possesses. 

In  Argentina  the  five-year  records  at  Cordoba-Pilar  have 
furnished  the  following  system  of  monthly  corrections,  which, 
added  to  the  zenith  value  of  the  intensity,  produce  results  such 
as  are  illustrated  in  the  preceding  diagrams.  The  necessity  for 
these  corrections  is  evident  from  the  fact  that  the  normal  in- 
tensity of  radiation  is  always  lessened  by  atmospheric  impurities, 
the  changes  being  in  one  direction.  Hence,  any  mean  value  of  a 
series  depends  upon  the  number  of  imperfect  days  which  have 
been  incorporated,  so  that  the  resulting  mean  values  merely 
reflect  the  effects  of  the  admixture  of  such  defective  conditions. 
The  following  tables  indicate  the  corrections  referred  to,  with 
examples  of  the  results.  It  is  certainly  improper  to  ascribe  to 
the  radiation  of  the  sun  itself  those  terms  which  depend  upon  the 
local  atmospheric  conditions. 


370 


A   TREATISE   ON   THE    SUN'S    RADIATION 


TABLE  115 

CORDOBA-PlLAR,    1912-1917 

Table  of  Reduction  to  the  Mean  Clear  System 


Month 

Clear 

Haze 

Dust 

January. 

-0  010 

+0  040 

+0  170 

February  

+0  .  030 

+0  .  050 

+0.220 

March  
April  .  . 

-0.030 
-0  030 

+0.060 
+0  060 

+0.150 
+0  150 

May  

+0.020 

+0  .  050 

+0.200 

June  

+0.030 

+0.060 

+0.210 

July  .  . 

+0  020 

+0  050 

+0.200 

August  
September  

+0.010 
-0.010 

+0.020 
0.000 

+0.190 
+0.170 

October 

—0  020 

—0  010 

+0  160 

November  
December  

-0.030 
+0.030 

0.000 
+0.020 

+0.150 
+0.210 

TABLE  116 
DATA  REDUCED  TO  THE  MEAN  CLEAR  DAYS 


Months 

1912 

1913 

1914 

1915 

1916 

1917 

Jan.,     Feb 

4  059 

3  946 

4  001 

4  012 

3  888 

3  960 

Mar.,   April  . 

4.013 

4.031 

3.921 

3  949 

3.876 

3  957 

May,    June  
Tulv,    Auc 

3.988 
4  037 

4.007 
3  995 

4.022 
3  979 

4.033 
3  952 

3.920 
3  974 

3.956 
3  952 

Sept.,  Oct  
Nov.,   Dec  

3.988 
3.934 

4.003 
4.076 

4.030 
4.029 

3.959 
4.023 

3.886 
3.973 

Means 

4  004 

4  014 

3  997 

3  988 

3  913 

3  956 

It  is  seen,  among  other  things,  that  the  year  1912  is  near 
the  maximum,  while  it  was  recorded  and  discussed  in  the  Northern 
Hemisphere — Mt.  Wilson,  Mt.  Weather,  Washington,  Bassour — 
as  a  minimum,  the  latter  being  due  simply  to  the  dust  in  the 
atmosphere  from  the  volcano  Katmai.  In  Pilar,  during  dry 
weather,  a  wind  storm  fills  the  lower  strata  with  dust,  and  the 
observed  intensity  drops  to  3.70,  3.60,  or  even  3.40  calories. 
Hence,  the  inclusion  of  such  dust  effects  in  any  mean  values 
simply  vitiates  them,  and  the  results  are  useless.  Until  these 
defects  can  be  fully  eliminated  from  the  observed  radiation 


SYSTEMS   OF   UNITS   EMPLOYED   IN   METEOROLOGY  371 

intensities,   the  data  can  not  be  utilized  in  any  of  the  solar 
radiation  problems. 

The  Systems  of  Units  Employed  in  Meteorology 

In  the  United  States  the  public  use  of  the  British  system  of 
measures  made  it  necessary  to  employ  some  of  them  in  the 
Government  Weather  Service.  These  are  the  pressure  in  inches 
of  mercury,  the  temperature  in  degrees  Fahrenheit,  the  wind 
velocity  in  miles  per  hour.  The  density  has  been  always  er- 
roneously computed  on  the  adiabatic  system,  the  gas  efficiency 
has  never  been  other  than  a  constant,  and  all  the  dependent 
thermodynamic  terms  are  consequently  in  error.  These  weather- 
map  data  are,  therefore,  inconsistent,  and  can  be  reduced  to  the 
M.  K.  S.  or  the  C.  G.  S.  systems  only  by  laborious  transforma- 
tions, which  it  is  impractical  to  apply  to  so  large  a  mass  of 
records  as  has  been  compiled  in  the  United  States  since  1871. 
These  data  are,  therefore,  only  remotely  accessible  to  scientific 
studies.  In  order  to  supplement  this  public  system  the  Weather 
Bureau  has  employed  a  different  but  mixed  system,  compiled 
from  several  sources.  Similarly,  the  International  system  is  not 
self -consistent,  and  is  likewise  inconvenient  in  scientific  researches. 
Furthermore,  these  imperfect  data  are  assembled  in  such  a  way 
as  further  to  embarrass  their  value. 

1.  The  Washington  8  A.M.  and  8  P.M.  hours  are  made  the 
times   for   simultaneous   observations   throughout   the   United 
States,  but  they  become  7  A.M.,  7  P.M.  at  Chicago,  6  A.M.,  6  P.M. 
at  Omaha,  and  5  A.M.,  5  P.M.  at  San  Francisco,  so  that  the  daily 
means  are  always  incomplete  and  generally  erroneous. 

2.  These   daily   means   of   inconsistent   data   are   compiled 
in  monthly  tables,  according  to  the  calendar  months,  but  these 
have  no  proper  connection  with  solar  recurrent  phenomena. 
It  would  be  difficult  to  devise  a  program  of  work  more  com- 
petent than  that  one  to  obscure  the  fact  of  solar  synchronism 
in  the  terrestrial  weather  conditions.     Table  117  contains  an 
example  of  these  systems,  based  upon  the  balloon  ascension  at 
Drexel,    Nebraska,    October    31,    1915,    which    illustrates    the 
divergent  and  inconsistent  methods  in  use. 


372 


A   TREATISE    ON    THE    SUN  S    RADIATION 


3 


§ 

PQ 
i> 

i-H 

*•<        W 

31 

PQ     2 


s 


•sto 


wg 


cj        .      a; 

&>     G     o> 

111 

0°          S  ' 


»—  1        CO        T-I        CO  l>  OS        i—  1        O5 


•3 


4J        C3 


O       fn       ^       O 
CO  I-H 


a  g 

CO       i-l 


<u 

"O 

.&  J  2 

c    e  s 

8p  o> 

e  c 


s 


I 

8 

:     o 


x 


rt    0 


o 
° 


i 


°.       00  . 

'Td^       00  ^^  r7(       ^5       ^D 

00<M  0  °9OOO          w. 

CMT^  (M  O5rHOi              rH 


OJ  ,.., 


fe«          I 
N 


3 

,  -U 

•     £     2 

5    i    8, 


o    -g    - 

" 


1     -8 


b/o 

S 

W 


5 


THE  BAR  AS  A  PRACTICAL  UNIT  FOR  ABSOLUTE  PRESSURE  373 

The  Bar  as  a  Practical  Unit  for  Absolute  Pressure 

An  effort  has  been  made  to  introduce  a  new  practical  unit  of 
absolute  pressure  into  meteorology,  depending  upon  1,000,000 
C.  G.  S.  units  of  pressure,  as  follows: 


C.  G.  S. 

McAdie 

von  Bjerknes 

Bigelow 

1000000  
1000 

1  megabar 
1  kilobar 

1  bar 
1  millibar 

1  kilobar 
1  bar 

1  

1  bar 

1  microbar 

1  millibar 

McAdie*  begins  with  1  bar  =  1  C.  G.  S.  unit  or  dyne;  and 
proceeds  through  1  kilobar  =  1,000  C.  G.  S.,  to  1  megabar  = 
1,000,000  C.  G.  S.  On  the  other  hand,  von  Bjerknes  proceeds 
in  the  opposite  direction,  from  1  bar  =  1,000,000  C.  G.  S., 
through  1  millibar  =  1,000  C.  G.  S.  to  1  microbar  =  1  C.  G.  S. 
unit.  But  it  is  evident  that,  in  practise,  the  natural  unit  of 
pressure  is  1,000  C.  G.  5.,  and  it  becomes  1  kilobar  =  1,000,000 
C.  G.  S.  and  1  millibar  =  1  C.  G.  S.  Pressure  would  then  be 
described  in  convenient  language,  for  example: 


C.  G.  S. 

1013.220 
955.180 
900.100 
848.140 
798.930 


M.  K.  S. 

1013.22 

955.18 
900.10 
848.14 
798.93 


Bars 


This  conforms  closely  to  the  customary  nomenclature  of 
millimeters  of  mercury  in  its  outward  form,  and  they  become 
nearly  identical  using  the  factor  of  multiplication  0.75.  Compare 
"Meteorological  Treatise,"  p.  4.  The  introduction  of  practical 
units  obscures  the  true  significance  of  the  data  in  higher  meteor- 
ology, and  in  computations  of  the  thermodynamic  and  radiation 
formulas  only  the  simple  M.  K.  S.  and  C.  G.  S.  systems  can  be 
utilized.  In  solar  thermodynamics  these  simplifications  are 
wholly  impossible.  The  M.  K.  S.  system  is  convenient  for 


"  The  Principles  of  Aerography,"  p.  30,  1917. 


374  A   TREATISE    ON    THE    SUN'S    RADIATION 

General   Meteorology,  and  the  C.  G.  S.  system   is   better    for 
atomic  physics.     No  mixed  systems  of  any  sort  are  permissible. 

The  Long  Range  Forecasts  of  Weather  Conditions 

In  view  of  the  immense  possibilities  of  this  branch  of  solar 
physics,  which  is  still  undeveloped,  and  the  fact  that  there  is 
a  persistent  synchronism  between  solar  and  terrestrial  atmos- 
pheric phenomena,  already  traced  for  more  than  half  a  century, 
it  is  desirable  that  meteorologists  should  generally  arrange  their 
data  in  a  form  for  immediate  and  convenient  use.  At  present 
the  observations  are  made  solely  for  the  construction  of  syn- 
chronous weather  charts,  in  relative  systems  of  pressure,  tem- 
perature, and  wind  velocity,  but  without  regard  to  the  general 
requirements  of  science.  The  mass  of  such  observations  has 
become  so  enormous  that  it  is  impractical  to  continue  to  make 
the  necessary  transformations.  The  following  reforms  are 
recommended  to  all  meteorological  services: 

1.  Adoption  of  the  (M.  K.  S.)  absolute  system  for  pressure, 
temperature,  and  wind  vectors. 

2.  Observations  at  the  local  hours  primarily,  but  reduced  to 
the  synoptic  chart  hours  for  weather  forecasts. 

3.  The  assembling  of  observations  in  the  26.68-day  solar 
period  and  the  abandonment  of  the  calendar-month  system. 

4.  The  closest  alliance  between  astrophysicists  and  meteor- 
ologists. 

5.  Extension  of  the  non-adiabatic   thermodynamics  to  all 
branches  of  atmospheric  physics. 

The  current  methods  of  operation  will  continue  to  conceal 
and  distort  all  kinds  of  solar  effects  in  the  terrestrial  processes 
merely  in  consequence  of  the  bad  workmanship.  Until  these 
steps  in  reform  have  been  taken,  all  critical  speculations  and 
skeptical  remarks  based  upon  inaccurate  data  and  methods  are 
useless,  as  they  hinder  the  progress  of  science. 


APPENDIX 

The  Pyrheliometer  and  the  Poynting  Equation 

Heaviside  remarks  that  the  Poynting  equation  constitutes 
the  first  proof  of  the  truth  of  Maxwell's  electromagnetic  theory  of 
radiation,  and  it  follows  that  every  apparatus  intended  to 
measure  the  intensity  of  the  solar  radiation — pyrheliometer, 
bolometer — must  conform  in  its  theory  to  that  equation.  It 
is  written: 


and  it  means  that  during  unsteady  temperatures  the  surface 
flux  of  the  radiation,  as  received  on  the  instrument,  is  equivalent 
to  the  volume  density  absorbed  within  the  apparatus,  plus  the 
free  heat  or  waste  due  to  lack  of  equilibrium  while  the  tempera- 
ture is  adjusting  itself  to  that  of  the  corresponding  black  radia- 
tion. During  such  an  adjustment  there  is  a  transformation  of  the 
molecular  and  atomic  forces  within  the  apparatus,  by  means  of 
the  kinetic  and  the  potential  energies  through  complex  inter- 
changes which  are  difficult  at  present  to  follow.  The  bolometer 
is  not  an  absolute  instrument,  but  merely  measures  relative 
ordinates  in  the  spectrum,  which  indicate  the  efficient  tempera- 
ture of  the  radiation  indirectly.  The  pyrheliometer  measures 
only  the  rate  of  change  in  a  temperature  thermometer,  such  as 
the  varying  A  produces,  but  it  does  not  distinguish  clearly 
either  the  kinetic  energy,  the  potential  energy,  or  the  free  heat, 
from  one  another.  The  kinetic  energy,  the  potential  energy, 
the  work  of  expansion  energy,  and  the  free  heat,  are  all  in- 
volved in  the  right  side  of  the  Poynting  equation,  after  the 
electromagnetic  energy  has  been  transformed  into  the  equivalent 
thermodynamic  energies.  Since  the  pyrheliometer  measures 

375 


376  APPENDIX 

only  the  relative  changes  of  temperature  in  the  unit  time  at 
different  zenith  distances,  and  not  the  absolute  volume  density 
in  the  final  state  at  any  time,  it  follows  that  the  Langley-Abbot 
method  of  discussion,  which  seeks  to  substitute  the  Bouguer 
formula  of  depletion  for  the  Poynting  equation  of  equilibrium,  is 
wholly  misleading.  When  the  spectrum  of  6,000°  is  arbitrarily 
selected  as  that  to  which  the  pyrheliometer  is  to  be  referred, 
this  is  contradicted  by  the  bolometer  ordinates;  when  the 
extrapolation  is  made  from  the  surface  to  the  vanishing  plane, 
from  sec.  z  =  1  to  sec.  2  =  0,  the  Bouguer  formula  is  abandoned; 
when  the  relative  intensity  of  the  observed  radiation  is  as- 
sumed to  be  complete,  or  absolute  in  amount,  it  is  evident 
that  this  is  falsely  identified  with  the  free  heat  implied  in  A, 
and  that  the  potential  energy  is  specifically  neglected.  The 
following  three  pieces  of  apparatus  have  done  great  injury  to 
meteorology:  the  mercurial  barometer,  by  substituting  some 
arbitrary  scale  value  of  pressure  for  the  absolute  units  of  force; 
the  evaporation  pan,  by  its  erroneous  distortion  of  the  amount 
of  water  lost  as  compared  with  a  large  free  water  surface;  and 
the  pyrheliometer,  by  its  incorrect  interpretation  of  the  solar 
constant  of  radiation,  as  compared  with  the  complete  theory  of 
the  Poynting  equation,  which  is  fully  confirmed  by  the  results 
of  the  thermodynamics  of  the  terrestrial  atmosphere,  of  the 
solar  atmosphere  and  of  the  bolometric  spectra.  There  is  prob- 
ably no  apparatus  more  difficult  to  interpret  correctly  than  is  the 
pyrheliometer,  because  it  demands  a  full  knowledge  of  the 
behavior  of  radiation  in  gases,  in  glass,  in  mercury,  in  metals, 
during  variable  transformations,  in  which  the  kinetic,  potential, 
expansion  and  free  heat  energies  are  all  undergoing  mutual 
readjustments.  To  compare  it  with  other  similar  apparatus 
is  to  obtain  only  relative  values  of  radiation,  and  to  ascribe 
these  to  the  absolute  solar  intensities  is  erroneous. 


INDEX 


Abbot,  C.  G.,  criticism  of  the  height  of  the  half-integral,  204 

method  of  approaching  the  problem  of  the  solar  constant,  193 

new  evidences,  194 

origin  of  the  sun-spots,  253 

theory  of  the  sharp  edge  of  the  sun,  79 

value  of  the  solar  constant,  7 
Absorbed  energy  in  solar  atmospheres,  118 
Acceleration  to  central  charge,  349,  352,  353,  354 
Activity,  29 

Adams,  origin  of  the  sun-spots,  253 
Adiabatic  equations,  11,  16 

gradients,  21,  53 

and  non-adiabaric  values  of  -r  =-,  318 

ratio  to  non-adiabatic  h,  332 
Angular  velocity,  352 
Anomalous  dispersion,  Julius,  79 
Argument,  resume  of,  240 
Association,  of  solar  elements,  8  63 
Astronomical  constants,  41 
Atmospheric  electricity : 

diurnal  convection,  342,  345,  346 

formulas,  331 

solar,  339,  341 

terrestrial,  333,  340 
Atmospheric  pressure,  313,  347 

bar  as  a  unit,  373 
Atomic  weights,  56 
Auroras,  cause  of,  339 

B 

Balloon  ascension,  Uccle,  September  13,  1911,  13,  332,  334, 

bar  as  unit  of  pressure,  373 
Balmer's  formula,  351 
Bigelow,  equations  in  atmospheres,  18 

formulas,  324 

functions  for  the  potential  energy  h,  307 

nuclear  charge  and  radius  in  atoms,  354 

origin  of  sun-spots,  253 

theory  of  radiation  by  collisions,  304,  363 
Bjerknes,  bar  as  unit  of  pressure,  372 
Black  radiation,  elements  of,  168 
Bohr,  theory  of  non-radiating  orbits,  304,  350,  351 
Bolometer,  13 

spectrum,  84,  375 

Boltzmann,  entropy  coefficient,  3,  142,  145,  166,  348 
Bouguer,  formula  of  depletion,  83,  115 
Boyle-Gay  Lussac  Law,  321 

in  the  terrestrial  forms,  1,  17,  347 

377 


378  INDEX 


Calcium,  double  reversal  lines,  246 

Carbon,  as  monatomic,  20 

Character  numbers,  international,  366 

Charge,  electric  on  surface  of  the  sun,  9,  342 

Check,  on  the  computations,  160 

Chemical  elements,  height  of,  82 

Chromosphere,  268 

Circulation,  general,  of  the  sun  in  latitude,  265 

general,  of  the  sun  in  longitude,  267 

relation  to  the  terrestrial  magnetic  fields,  269 

rise  and  fall  of  a  zonal  vortex,  265 
Clear  day,  mean  clear  day  system,  370 
Coefficients,  distribution  of,  99 

in  the  Stefan  Law,  28 

mean  values,  104 

of  radiation,  113 

variable,  3,  17,  27,  132 

variable  in  the  Wien-Planck  Law,  128 
Constants,  4,  19,  22 

hyperbolic  law  of  pressure,  68 

hyperbolic  law  of  temperature,  62 

solar  of  radiation,  103 

Wien-Planck  Law,  133,  312,  348 
Constituents  of  radiation,  40 

Convergence  number,  Bigelow  and  Bohr,  351,  353,  354 
Conversion  factors,  surface-flux,  2 

thermodynamic  term,  121 

volume-density,  2 

Cordoba-Pilar  data  of  atmospheric  electricity,  343 
Corona,  solar  inner,  268 

solar  outer,  269 

solar  polar  rays,  273 
Coronal  strata,  209 
Curl,  general  laws  of,  29 


Day,  A.  L.,  volume  density  of  radiation,  2,  122 
Decay,  exponential  law  of,  173 
Density,  321,  348 

Bessel's  formula,  289 

in  the  numbers  n  and  N,  313 
Depletion,  logarithmic  law  of,  109 

successive  steps  from  5.85  to  1.50  calories,  219 
Derivation  of  orbital  formulas,  349 
Dimensions,  coefficient  in  the  Stefan  Law,  122 

equation  of  condition,  2 

surface-flux  of  radiation,  2 

systems  (M.K.S.)  and  (C.G.S.),  120 

terms,  38 

volume-density  of  radiation,  2 
Discontinuity,  in  the  radiation  intensity,  191 
Dissociation,  hydrogen,  103 

solar  elements,  7,  63 
Distributions,  by  the  hyperbolic  law,  62 
Disturbances  of  magnetic  field,  339 
Diurnal  components,  343 
Divergence,  29 


INDEX  379 

Drexel,  observations  of  electricity  in  atmosphere,  339,  340 

example  of  units,  371 
Dust,  effect  of  upon  the  intensity  of  radiation,  368 

£ 

Effect  of  dust,  368 
Efficiency,  321,  322,  347,  348 

factor,  199 

in  N  k,  321 

Einstein's  Law,  331,  333 

Electric  orbits  in  the  K  and  L  series,  358-361 
Electric  potential,  surface,  342,  343 
Electromagnetic  field,  general  formulas,  28,  363 

six  subdivisions  in  the  earth's  atmosphere,  224 
Elements  of  black  radiation,  168 
Emission,  temperature  of,  34 
Energy,  absorbed  in  the  solar  atmosphere,  117 

black  radiation,  177 

conservation  of,  90 

dissipated,  31 

distribution  in  the  earth's  atmosphere,  214 

distribution  of  inner,  92 

electromagnetic  kinetic,  28,  212 

electromagnetic  potential,  28,  213 

inner,  12 

kinetic,  external,  and  internal,  12,  13,  19 

potential,  external,  and  internal,  12 
Entropy  coefficients,  Boltzmann's,  3,  145 

distribution  of,  88 

formulas,  27,  35 

specific,  181  m 
Equation  of  condition,  %,  12,  16 

adiabatic,  16 

non-adiabatic,  16 

Evaluation  of  the  series  factors,  357 
Example  of  the  dust  effect,  369 
Exponent  in  the  Stefan  Law,  27 

distribution  of,  100 

mean  values,  104 

F 

Factor  of  efficiency,  197 

evaluation  of  the  series,  357 

of  transformation  from  the  earth  to  the  sun,  21 
Faculse,  movement  in  the  11-year  period,  263 

solar,  8,  247,  250 
Faye,  origin  of  the  sun-spots,  253 
Filamentary  structure  of  the  solar  surface,  249 
First  law  of  thermodynamics,  27,  90 
Flocculi,  solar,  8,  247,  250 
Flux,  mean,  33 

Poynting's,  15,  30,  31,  123-126 
Forecast,  annual  long  range,  236 

long  range  in  general,  374 

short-  and  long-range,  237 
Formulas,  resume  of  thermodynamic,  201 
Fox,  origin  of  sun-spots,  252 
Free  heat,  12,  15,  27,  86-88,  323 
Frequency,  relation  to  ionization  and  potential  energy,  330,  332,  334-339 


380  INDEX 

G 

Gas  efficiency,  15,  16,  78,  128,  347 
Gradients,  long  and  short  vertical,  256 
Granulation,  relation  to  other  phenomena,  245 

solar,  8,  244,  247,  249 
Gravity  acceleration,  15 


Hale,  G.  E.,  origin  of  the  sun-spots,  253 

double  reversal  lines,  247 

Zeeman  effect,  272 

Heaviside,  examples  of  electromagnetic  energy,  33,  363 
Height  of  atmosphere,  effective  thermodynamic,  207 
Herschel,  origin  of  the  sun-spots,  251 
Historical  remarks,  9 
Hydrogen,  double  reversal  lines,  247 

monatomic  and  diatomic,  20,  63,  102 
Hydrostatic  pressure,  15,  322,  347 
Hyperbolic  law  of  distribution,  62 


Index  of  refraction,  terrestrial,  294 

solar,  294 
Inner  energy,  15,  27,  117 

in  gases,  310,  323 

Institutes,  argument  for  four  international,  237 
Intensity  of  radiation,  35 

of  the  entropy  of  radiation,  181 

variable  solar,  211,  364^367 
International  character  numbers,  366 
Interpenetration  of  electronic  orbits,  362 
lonization,  and  atmospheric  electricity,  329,  330,  334-339 

association  with  changes  in  temperature,  230 

in  the  upper  strata,  225,  229 

relative  to  potential  energy  and  frequency,  330 
Isothermal,  gradients,  52 

layer,  59,  60 

layer,  solar,  5,  8,  338 

layer,  terrestrial,  8,  337 

mean  temperature,  66 
Isothermal  shell,  effect  upon  other  solar  phenomena,  242 

relation  to  the  origin  of  sun-spot  vortices,  258 


Joule's  heat,  29 

Julius,  anomalous  dispersion,  79 
origin  of  sun-spots,  252 


Kinetic  energy,  in  gases,  310 

in  gram  calories  centimeter-square  minute,  257 

in  radiation,  from  thermodynamics,  321 

mean  values,  141 

of  solar  radiation,  15 

one  molecule  E0  as  a  variable,  14Q 


INDEX  381 


Kinetic  energy  per  unit  volume,  159 

relation  to  potential  energy  in  orbits,  349 
universal  mean,  134 

Kirchhoff's  Law,  176 

K-radiation  series,  353,  358,  359 

Kurlbaum  coefficient,  84,  113,  123 


Langley,  origin  of  sun-spots,  252 

Langley- Abbot,  method  of  determining  the  solar  constant,  115 

Lightning,  343 

Limb  of  the  sun,  sharp  edge,  79 

Line  and  band,  absorbents,  222 

scattering  and  absorption,  Fowle,  222 
Lockyer,  origin  of  the  sun-spots,  252 
Logarithmic,  law  of  depletion,  109 

spiral  for  oc  =  35°  10',  111 
Long  range  forecasts,  365 

Lorentz,  volume  density  of  radiation,  2,  122,  124 
L-radiation  series,  353,  358,  361 

M 

Magnetic  field,  relation  to  the  solar  circulation,  267 

generation  in  the  sun,  29 
Magnetism,  solar,  271 
Mass  of  hydrogen  atom,  140 
Maxwell's  Law  of  circulation,  29 
McAdie,  bar  as  unit  of  pressure,  373 
Mean  kinetic  energy,  143 
Mechanical  force,  36,  179 
Meteorological  treatise,  1,  10 
Meteorology,  systems  01  units  in,  371-372 
Millikan,  nuclear  charge  and  radius  in  atoms,  356 
Mirror — wall  enclosure,  204 
Molecular  potential  energy,  terrestrial,  325,  349 

potential  energy,  solar,  326 

weights,  56,  57 
Molecules,  half  atomic  weight  number,  312 

number  per  cu.  cm.,  136 

number  per  gram,  139 
Moseley's  Law,  355,  356 

N 

Non-adiabatic  equations,  10,  16 

gradients,  52 

to  adiabatic  ratio  for  h,  332 
Nuclear  atomic  charges,  352,  353,  354 
Number  of  molecules  in  gases,  311 

per  cu.  cm.,  136 

per  gram,  139 

variable  N',  140 
Numbers  atomic,  359,  360,  361 

O 

Oppolzer,  origin  of  sun-spots,  252 
Orbital  formulas,  derivation  of,  352 
Orbits,  interpenetration  of  electronic,  361 


382  INDEX 


Photosphere,  definition  by  the  pressure,  53,  68 
Planck,  equations  in  the  aether,  18 

formulas,  35,  324 

on  Kirchhoff's  Law,  123 

theory  of  oscillators,  304 

volume  density  of  radiation,  2,  123 

Wirkungsquantum,  17,  146 

Wirkungsquantum  in  the  thermodynamic  form,  311,  325 
Polarity,  magnetic  in  sun-spots,  278 
Potential  energy,  constants,  6,  12,  331 

distribution  of,  70 

electric  in  the  atmosphere,  340-347 

in  the  earth's  atmosphere,  214,  325 

of  solar  radiation,  15,  27,  210,  339 

relation  to  ionization  and  frequency,  331,  334-339 

relation  to  kinetic  energy  in  orbits,  349 

thermodynamic  radiation,  321 
Poynting's  equation,  30,  170,  192,  375 
Pressure,  18,  321,  347,  348 

in  gases,  311 

of  light,  178 

of  radiation,  32,  35,  176 

ratio  of  thermodynamic  to  the  radiation,  186 
Product,  scalar,  29 

vector,  29 
Prominences,  formation  of,  250 

movement  in  the  11-year  period,  263 

solar,  8 
Pyrheliometer,  direct  readings  to  22,000  meters,  225 

Bigelow's  method  of  reduction,  226 

elimination  of  the  dust  effect,  368-369 

Poynting's  equation,  375 

reduction  to  3.98  calories  without  extrapolation,  228 


Radiation,  absorbed,  5 

annual  values,  232,  369 

as  a  radioactive  discharge,  7,  8 

black  solar,  197 

constituents  of,  224 

curves  of  extinction  in  the  sun's  atmosphere,  196 

effective  at  the  earth,  7,  195,  199 

entropy,  35 

free,  7 

function,  324 

hs  (Bigelow),  hP  (Planck),  314 

potential  energy  of,  211 

scattered,  7 

solar  constant  of,  7,  103,  221 

summary  in  the  earth's  and  sun's  atmospheres,  189 

summary  in  the  1000-meter  strata,  207 

summary  of  absorbed  and  black,  205 

three  theories  of,  304 

variable  solar,  211,  364-367 
Radiation  potential,  coefficients  and  exponents  of,  96 

distribution  of,  95 


INDEX  383 

Radiation  stratum,  depth  of  the,  104 

Ratio  of  the  non-adiabatic  to  the  adkbatic  values  of  h,  317 

Refraction,  circular,  77 

atmospheric  and  scattering,  285 

formulas  of,  286 

index  of  solar,  294 

index  of  terrestrial,  296 

Relations  between  ionization,  potential  energy  and  frequency,  330,  334-338 
Resume  of  thermodynamic  results,  216 
Reversal  of  spectral  lines,  double,  247 
Richardson,  volume-density  of  radiation,  122 


Saltum,  change  in  the  coefficients  and  exponents,  97 
Sanford,  X-radiation  and  L-radiation,  355,  359,  360,  361 
Scattered  radiation,  mode  of  production,  250 

atmospheric,  285 

Schaeberle,  theory  of  sun-spots,  252 
Schmidt's  theory  of  the  sharp  limb  of  the  solar  disk,  79 

theory  of  sun-spots,  252 
Secchi,  theory  of  sun-spots,  252 
Second  Law  of  thermodynamics,  27,  93 
Semidiurnal  periods,  origin,  345 
Series  of  spectral  lines,  351 

Sheath,  cooling  on  the  edges  of  large  masses,  248 
Sidgreaves,  theory  of  sun-spots,  253 
Solar  constant,  Abbot's  value  of,  4,  67 

Bigelow's  value,  4 

of  radiation,  4,  115,  103 

summary  of  results  for  3.98  calories,  211 
Sola'-  radiation,  general  distribution  of,  116 

origin  of,  6,  112 
Specific  heat,  19 

in  gases,  310 

Specific  kinetic  energy,  322 
Spectra,  characteristics  of  the  solar,  281 

principal  lines  of,  282 

Spectral  lines  in  series,  351,  353,  354,  357-361 
Spectrum,  distribution  of,  337 

flash,  8 

formulas,  35 

high  and  low  temperature,  8 
Stark  effect,  342 
Stefan  Law,  6,  34,  84 

variable  coefficients  in,  120 
Stratum,  depth  of  the  radiation,  104 
Summary,  328 
Sun,  a  black  radiator,  199 

a  magnetized  sphere,  272 
Sun-spots,  8 

inward  and  outward  velocities,  261 

movement  in  the  11-year  period,  263 

origin  of,  251 

polarity  by  the  Zeeman  effect,  277 
Synchronism,  solar  and  terrestrial,  368 
Systems  of  units,  371-372 


384  INDEX 


Temperature,  dimensions,  17,  67 

distribution  of,  58,  59 

emission,  34 

examples  of  vertical  changes,  246 

heights  of  the  same  values,  62 

isothermal  shell,  8 

mean  of  the  radiation,  114 

on  the  same  level,  64 

radiation,  4 

Thermal  coefficient,  138 
Thermodynamic  data,  method  for  annual  values,  232 

formulas  and  the  spectrum,  329 

processes  invisible  and  deep-seated,  262 

resume  of  formulas,  201 

summary  of  terrestrial,  203 

terms  in  practical  forms,  348 

values"  of  hB  (I),  313 

values  of  hp,  313 

values  of  hB  (II),  320 
Thermodynamics,  First  law  of,  2 
Thunderstorms,  origin,  343 
Total  inner  energy,  323 
Transmission  coefficients,  291-293 
Trials,  method  of,  43 

U 

Uhler,  X-radiation  and  L-radiation,  359,  360,  361 
Units,  systems  employed  in  meteorology,  371-372 
Unpolarized  intensity  of  radiation,  185 


Vanishing  plane,  solar,  2 
Variable,  k  in  all  atmospheres,  312 

potential  coefficient  h,  313 
Variations  of  the  solar  radiation  in  the  26.68-day  period,  365 

radiation  in  the  365-day  period,  367 
Velocities,  inward  and  outward  in  the  sun-spots,  261 
Velocity,  arithmetical  mean,  131 

mean  square,  132 

Very,  F.  W.,  erroneous  criticism,  122,  160,  192 
Voltage  of  atmospheric  electricity,  340-343 
Volume,  19,  322,  330,  348 

density  of  radiation,  314-316 

density,  specific,  170 

intensity  coefficient,  154 

specific  and  molecular,  131 
Vortex,  of  the  general  circulation,  265 


W 

Weather  conditions,  long  range  forecasts,  374 

Weights,  atomic  and  molecular,  56 

Wien,  W.,  displacement  formula,  variations  in,  319,  355 

volume  density  of  radiation,  2 
Wien- Planck,  coefficients  in  law  (clt  cz),  17,  149-152 


INDEX  385 


Wien-Planck  formula,  218 

formula  for  black  radiation,  305 
second  computation,  163 
variable  coefficients,  120,  127 
Wirkungsquantum,  17,  145,  212 
Work,  external,  12,  15,  19,  27,  322,  323 


Young,  origin  of  the  sun-spots,  252 


Zeeman  effect,  9 

formulas  of,  276 
identification  in  sun-spots,  272 
polarity  in  sun-spots,  277 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


ASTRQNDMY   LIBRARY 


jww-rrrast 


DCTr-aG   1062 


Due  end  of  WINTER  quartw 
Subject  to  recall  after 


LD  21-100m-ll,'49(B7146sl6)476 


U.C.  BERKELEY  LIBRARIES 


' 


